Block coding into 24 coordinates and detection of transmitted signals
This invention relates to means and a method of signalling binary or other numeric data on a communications channel. The means and method encode blocks of b units of data into 24 coordinate signal values which 24 coordinate values are then modulated on a carrier by whatever modulation is desired. The means and method of the invention also extend to the detection and decoding at the receiver end of the channel. 
     By the term `numeric data` in the application and claims there is envisaged data in binary or other numeric forms which will be supplied by physical signals in groups for conversion to message points. In the great majority of cases such data will be represented by binary digits. However, it is noted that the invention extends to data in the form of other numeric bases. 
     As hereinafter discussed in more detail, the invention particularly relates to a novel means and method of detecting the 24 coordinates after transmission and demodulation but before decoding. 
     The overall encoding, detecting and decoding means utilizes the fact that 24 independent signal values may be treated as the coordinates of a point in 24 dimensions. Signal structures in 24 dimensional space provide performance that is superior to other dimensional signal structures investigated. The superiority may be realized in terms of fewer errors at a given signalling rate or in terms of a higher signalling rate for a given statistical possibility of error. 
     A `message point` is the point in 24 dimensions defined by the 24 coordinates. 
     The term `message point` is not only used for points in 24 dimensions which are used for signalling but also for points in lesser dimensions. For example and as dealt with hereinafter a message point in H dimensions may be identified by the combination of a message point in F dimensions and a message point in G dimensions where F+G equal H. Thus, although the signalling is of message points in 24 dimensions, the term message point applies to points in less than 24 dimensions which are not signalled per se but which are used in algorithms related to encoding or decoding the 24 dimensional message point signalled. It may help with following the terminology herein to note that, in higher dimensions than one, a number of coordinates equal to the dimension is required to define a message point. In one dimension the coordinate is also the message point. 
     For encoding blocks of b data bits, signalling speed is related to the number of bits which may be encoded per block. A look-up table may be used for small values of b. However, the number of entires in such look up table varies as 2 b  so that for large values of b the requirements would exceed the capacity of known computer or microprocessor look up tables. 
     This invention provides means and a method for deriving the coordinates in 24 dimensions to identify individual blocks of b digits. In particular, algorithms are provided which allow the conversion of each block of b digits into message point coordinates in 24 space which coordinates uniquely define the block of b digits represented. 
     Generally the invention utilizes the fact that message points defined by 24 coordinates may be considered as located in concentric shells about the origin (the point which has 24 0 values for coordinates) or about another datum defined by 24 coordinates. 
     A shell therefore contains those message points where ##EQU1## is the same value where C1, C2, . . . C24 are the coordinates of the message point and CD1, CD2, . . . CD24 are the coordinates of the datum. 
     The invention generally provide means and method which involves treating each block of b bits as a number N which identifies the sequence of bits in the block. The simplest rule is that the number N is the binary number represented by the sequence of bits in the block. 
     The invention herein uses 24 coordinates encoded in accord with a Leech matrix. 
     A Leech matrix has a classical 12×24 form as demonstrated in Table 1 below: 
     
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Row No.                                                                   
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1        1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1                  
2        0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 1                  
3        0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1                  
4        0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0                  
5        0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 0 1                  
6        0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 1 0                  
7        0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 1 0 0                  
8        0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0 0                  
9        0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1                  
10       0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1                  
11       0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 1 1                  
12       0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 1 0                  
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     By adding all possible combination of rows of the Leech matrix column by column, modulo 2, (`modulo` is often abbreviated to `mod` herein) it is possible to produce 4096 unique 24 series of 0 and 1 values known as pattern vectors. An example the pattern vectors for the mod 2 sums for 0 rows and rows 1, 2, 3 are: 
     
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0 rows  000    000    000  000   000  000  000  000                       
1,2,3 rows                                                                
        111    000    000  000   000  011  011  001                       
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     Coordinates in accord with the following rules provide very good packing in 24 dimensions. (See Scientific American, January 1984, page 116 &#34;The packing of spheres, by N. J. A. Sloane)&#34;. 
     The rules for good packing in 24 dimensions are: 
     The 24 coordinates must be all even or all odd. 
     If even, the sum of the coordinates must be 0 mod 8; and 
     If odd, the sum of the coordinates must be 4 mod 8. 
     By good packing or relation to modulated communication signals is meant (a) that for given signalling power the message point defined by the 24 coordinates can have a relatively good chance (the best for known 24 space packing and better than other space packings) of being distinguished from the adjacent message points or, conversely (b) that one can use less power and have statistically a change of being distinguished from message points using other packings in 24 or other dimensions. 
     For encoding note the lattice of message points whose coordinates are all 0 mod 4 are centred about the point whose 24 coordinates are 0&#39;s and is related to the pattern vector with zeros in each of the 24 places. This lattice is known as the Z lattice. The remaining 4095 even and 4096 odd lattices are known as co-sets of the Z lattice are each respectively identified with one of the 4096 pattern vectors, and are reached by modifications of points on the Z lattice as hereinafter described. 
     Thus, in encoding the good packing is obtained with the Leech matrix by producing 24 coordinates on the Z lattice for modification in accord with a selected pattern vector. The modification is (when even coordinates are selected) to alter the Z lattice coordinates by 0 or 2 in the same sense depending on whether the selected pattern vector has a 1 or a 0 (and either but not both conventions may be used); and (when odd coordinates are selected) to alter the lattice coordinates by 1 in opposite senses depending on whether the selected pattern vector has a 1 or a 0 (and either but not both conventions may be used). (With odd coordinates, to maintain the modulo sum rule one coordinate must be increased or decreased by 3 instead of 1). 
     Thus a block of data bits may be converted into a number N. The number N is identified by the Leech pattern vector and whether consisting of even or odd coordinates together with a series of 24 coordinates. Selection of the vector and the choice of even or odd coordinates allows N to be divided by 8192 to give a quotient M and a remainder C. The quotient M is encoded as coordinates on the Z lattice (the lattice built upon the point whose coordinates are all 0&#39;s) as hereinafter described while twelve bits of C are used to selected the pattern vector to modify the Z lattice coordinates, and one bit of C indicates the selection of even or odd. The coordinates thus selected are converted to signals, modulated on a carrier and transmitted. On reception the demodulation produces values for the coordinates but these must be detected both as to quantum and as to which pattern vector was used and whether even or odd coordinates were used, before the most probable value of the transmitted coordinates may be selected and decoding performed by inversion of the encoding procedure. 
     This invention relates to the detection of 24 coordinate signals encoded as before described which utilizes a novel system of detecting at the receiver the pattern vector used in the transmission. The novel method of detection requires a (relatively) small number of decoding steps. 
     The same 4096 pattern vectors, which are developed from the conventional Leech matrix, may be developed from any matrix referred to as a `modified Leech matrix` or `modified matrix` derived from the conventional matrix, Table 1, by combinations of its rows added coordinate by coordinate mod 2, care being taken that the selection is such that the modified matrix will allow the development of the 4096 unique vectors. 
     The conventional or a modified Leech matrix may be modified by interchanging any whole columns. This will, of course, have the effect of correspondingly interchanging the pattern vector places and the effect on individual coordinates. This will not change the process described as long as the same matrix form is used both in encoding and decoding. 
     In the preferred embodiment as described herein the conventional Leech lattice has its columns interchanged and new rows made by the summation mod 2 of old rows (as referred to above and described in detail hereafter) to produce the matrix: 
     
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1       11111111      00000000 00000000                                   
2       00000000      11111111 00000000                                   
3       00000000      00000000 11111111                                   
4       00001111      00001111 00000000                                   
5       00110011      00110011 00000000                                   
6       01010101      01010101 00000000                                   
7       00000000      00001111 00001111                                   
8       00000000      00110011 00110011                                   
9       00000000      01010101 01010101                                   
10      01001101      01001101 01001101                                   
11      01010011      01010011 01010011                                   
12      01110100      01110100 01110100                                   
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     The modified matrix is then used in encoding by having the dictates of pattern vectors derived therefrom superposed on Z lattice coordinates. The modified coordinates are modulated on a carrier and transmitted. The received transmission is demodulated and detected in accord with the preferred method and means disclosed herein. The coordinates are then decoded to recover the number M and therefrom the number N by use of the number C. The preferred method of detection makes use of the modified matrix, to detect the selected pattern vector used and even or odd. These of course are used to produce the Z lattice coordinates used. 
     The invention is described in relation to encoding, modulation and transmission at the transmitter and reception demodulation detection and decoding at the receiver. Thereafter the preferred encode, detection and decode algorithms are set out. 
     In the encoding step the number of bits or of other numerical data is converted to the number N. This number is divided by 8196 to provide a quotient M and a remainder C. The 13 bits of remainder C identify the pattern vector of the modified Leech matrix and whether the coordinates will be even or odd. For the quotient M, a table is provided listing the numbers (not the coordinates) of suitable message points for a sequence of shells in 24 dimensions. The means provided determines the shell in the sequence whereat the total available points in the sequence is greatest without exceeding M. The value X 24  being M less the number of points in the sequence, identifies (without locating) a point in the next succeeding shell in the sequence which will define M. The coordinates corresponding to the point are obtained by the use of splitting algorithms whereby X 24  is identified by specified shells and corresponding values of X F  and X G  for each of F and G dimensions where F+G=H (H in the first `split`=24). By continuing the use of the splitting algorithm to lower and lower dimensional values, 24 corresponding values of the value X 24  in 1 dimension may be derived, the 24 one dimensional value identifying the value and sign of the coordinates C1, C2 . . . C24 which define the message point. These coordinates may then be modulated on the carrier. At the receiver, the demodulated and detected coordinates are subjected to a combining algorithm which is basically the reverse of the splitting algorithm and which allows the reconstruction of the number X 24  and the shell in 24 dimensions from the coordinates, and from that the numbers M and C (derived from the detection of the pattern vector and of whether even or odd was used) giving the number N and the block of b bits. 
     It will be noted that the use of splitting algorithms at the transmitter and of combining algorithms at the receiver avoid the use of large look up tables in converting the number M, identifying the block of b digits, to 24 message point coordinates for signalling and vice versa. Since look up tables of the required capacity are not available, the use of the splitting and combining algorithms allows signalling at speeds higher than previously. 
     Tables may be built up for use with the splitting algorithm beginning with the number of coordinates in 1 dimension which satisfy the coordinate requirements for signalling in 24 dimensions. These numbers are tabulated. It is found most convenient to tabulate these numbers using the concept of shells where each shell is numbered from 0 in regular intervals, each unit interval corresponding to an increase in the square of the radial distance, r 2  from the origin or datum in one dimension, by the square of the permissible interval between coordinates, the latter being used in integral form in most applications. 
     There is now discussed for 24 dimensions and using the modified Leech matrix (for use with the encoding algorithm to follow) an advantageous packing for encoding where the following rules apply. The coordinates are integers and must be all even or all odd. If all even the sum of the coordinate in the signalling dimension must be 0 modulo 8 and if all odd the sum of the coordinates must be 4 modulo 8. Using the Z lattice, the permissible coordinate interval is 4 and one (of 4096) pattern vectors provides such coordinates centred at the origin. The points on shells centered about the origin are contained on what is known as the &#34;Z lattice&#34;. Accordingly, in one dimension, the table of permissible values* tabulated by shells is: 
     
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Shell No.         No. of Available                                        
                               Integral                                   
(I or J) r.sup.2  Points       Coordinates                                
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S0        0       1            0                                          
S1       16       2            4 or -4                                    
S2       32       0            (none)                                     
S3       48       0            (none)                                     
S4       64       2            8 or -8                                    
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     and it will be noted that the Shell No is 1/16 the value of the radius squared. 
    
     The available message points in 2 dimensions may be derived using the summing algorithm ##EQU2## where n is the shell number, where the number of dimensions F plus the number of dimensions G equals the number of dimensions H; where V H , V F , V G  are the number of available (i.e. satisfying the coordinate requirements) points in H, F, G dimensions respectively and the intervals of I are constant and chosen to ensure that the sum includes all such available points up to n times the square of the interval. I indicates the shell number in dimension F while n-I (often referred to as J) is the shell number in dimension G. 
     With the algorithm and with F=1, G=1, H=2 the following table is provided for 2 dimensions. 
     
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Shell No.        Total Number of                                          
                              Integral                                    
(I or J) r.sup.2 Available Points                                         
                              Coordinates                                 
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S0        0      1            0,0                                         
S1       16      4            (±4, 0)(0, ±4)                        
S2       32      4            (±4, ±4)                              
S3       48      0            none                                        
S4       64      4            (±8, 0)(0, ±8)                        
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     It will be seen that the algorithm may be used to provide the numbers of available points for all values of H by applying the algorith to tables for F and G where F+G=H. In this way the available points in 24 D for the Z lattice may be determined. Tables of numbers of available coordinate values for dimensions 1, 2, 4, 6, 12, 24 for an origin centred (`Z`) lattice with intervals of 4 between coordinates are set out in tables Z1, Z2, Z4, Z6, Z12, `Offset` attached to the algorithms herein. It should be noted that the Z24 `Offset` table does not include the points which, although derived from the Z12 tables in accord with the algorithm, do not satisfy the modulo rules for the coordinate sum, even through such points would have been indicated by the algorithm. Thus, it will be noted that table Z24 omits the points which the algorithm would have provided for odd numbered shells, since these would not have satisfied the modulo sum rule that the coordinates must be zero modulo 8 (where the coordinates are all even as in the Z lattice). Thus the coordinates are determined on the Z lattice and modified in accord with the selection of a pattern vector and of even or odd coordinates. 
     A specific embodiment is now described. 
    
    
     FIG. 1 shows schematically the overall circuitry employed with this invention including the program instructions and algorithms employed therewith, 
     FIG. 2 shows the relationship of sequential uses of the splitting and combining algorithm. 
    
    
     In the drawings FIG. 1 schematically illustrates the functional operations performed in a communications system utilizing the invention. The functional operations are not intended to imply particular hardware or choices between hardware and software modes, except in blocks 20, 80 and 90. In block 20 and block 90 the microprocessors are programmed to perform the operation indicated. Although microprocessors are specified for use in various steps of the operation, the operation described may use any system adapted to provide the claimed means and methods. 
     Thus, as functionally illustrated in FIG. 1, serial binary data in blocks of b bits is scrambled at scrambler 15 and converted at serial to parallel convertor 10 into groups of b bits. At a data rate of 19,200 bits per second the data to be transmitted in one frame consists of 96 bits to be block encoded and if an auxiliary 200 bps channel is desired 97 bits must be block encoded. At 16,800 bits per second with an auxiliary channel, blocks of 85 bits must be encoded per block and at 14,400 bits per second, 73 bits must be encoded per block. 
     The groups of b bits are block encoded at block 20 into the values of 24 coordinates C1, C2, . . . C24 in accord with the encoding algorithm hereinafter provided. The values of the coordinates are converted to modulating signals in coordinate signal generate 30. The outputs of coordinate signal generator 30 and carrier generator 40 are converted at modulator 45 into a carrier modulated in accord with the values C1, C2, . . . C24. The invention is independent of the method of modulation. It will usually be preferred to use Quadrature Amplitude modulation QAM wherein the signals incorporating the coordinate values are modulated in twelve pairs by conventional quadrature methods. Several other forms of modulation are available. These include but are not limited to double side band-quadrature modulation (often abbreviated DSB-QAM). DSB-QAM includes modulation techniques such as phase-shift keying PSK, quadrature amplitude modulation (QAM), already referred to, and combined amplitude and phase modulation which have long been known in the art. 
     The modulated carrier signals from modulator 45 are provided to transmitter interface 50 and transmitted in the interfaced form to the channel. After reception from the channel at receiver interface 60 the received signals are demodulated, conditioned and equalized at block 70, all in accord with techniques well known to those skilled in the art. The output of block 70 is provided to the microprocessor detector and decoder 80 where by combined operation of the microprocessor and the detection method, the Leech pattern vector used and the transmitted coordinates C1, C2, . . . C24 are detected. The matrix in accord with the detection and decoding algorithms to be described converts these coordinates into the b bits of binary data which were supplied to the input of block 20 at the receiver. The reconstituted b bits are converted to serial binary data at convertor 90. The above cycle involving the encoding of b bits into 24 coordinate signals and resultant detection and decoding at the receiver will customarily be performed (at each end of the channel) 200 times per second, with the signalling speed in bits per second of bits b being determined by the number of bits b which are block encoded to correspond to each 24 coordinates. Circuitry for performing the functions described excepting those of blocks 20, and 80 is well known to those skilled in the art. 
     It is now proposed to describe the operation of the encoder 20, the description to be read with encoding algorithm appearing hereafter. 
     In the encoder 20 the b bits are to be encoded as a block of 24 coordinates so that dimension=24. Since a Leech pattern matrix is to be used, (see the Modified Leech matrix, used in the encoding algorithm which follows), there are 4096 pattern vectors which may be constructed therefrom by adding any combination of rows (including no rows) coordinate by coordinate modulo 2. The pattern matrix shown is not the classical form but is derived therefrom where each row of the pattern matrix is derived from a different combination (added coordinate by coordinate modulo 2) of the rows of the classical form including the corresponding row. The conventional Leech Matrix is shown below: 
     
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Row No.                                                                   
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1        1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1                  
2        0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 1                  
3        0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1                  
4        0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0                  
5        0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 0 1                  
6        0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 1 0                  
7        0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 1 0 0                  
8        0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0 0                  
9        0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1                  
10       0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1                  
11       0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 1 1                  
12       0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 1 0                  
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     To produce the modified matrix for preferred use in the encode and decode steps and the necessary matrix for the detection step, the columns of the above matrix are rearranged into the sequence. 
     1, 18, 24, 22, 19, 10, 23, 3, 6, 8, 21, 16, 12, 15, 17, 7, 9, 13, 2, 4, 14, 5, 20, 11. 
     And then by creating new rows from the sums (mod 2) of the following old rows: 
     
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New Row Old Rows                                                          
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1       1,3,10       (The `old rows` listed are                           
2       6,7,8,12     derived from a longer set                            
3       2,4,5,9,11   of mod 2 row additions                               
4       3,7,10,12    shown in Row Addition Table.)                        
5       3,7                                                               
6       3,7,8,10                                                          
7       5,7,11,12                                                         
8       2,4,7,11                                                          
9       4,5,7,8,11                                                        
10      3,5,7,8,10,11,12                                                  
11      3,4,7,8,11                                                        
12      2,4,5,8,10                                                        
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     to produce the modified Leech matrix as below: 
     
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1       11111111      00000000 00000000                                   
2       00000000      11111111 00000000                                   
3       00000000      00000000 11111111                                   
4       00001111      00001111 00000000                                   
5       00110011      00110011 00000000                                   
6       01010101      01010101 00000000                                   
7       00000000      00001111 00001111                                   
8       00000000      00110011 00110011                                   
9       00000000      01010101 01010101                                   
10      01001101      01001101 01001101                                   
11      01010011      01010011 01010011                                   
12      01110100      01110100 01110100                                   
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     (It may here be noted that the interchangeability of columns in a Leech matrix is merely one aspect of a more general rule that the coordinates of an orthogonal system may be interchanged freely as long as the interchanged order is maintained. Thus although coordinates in three dimensions are customarily written in the order (X,Y,Z,) they may be written (Z,X,Y,) (Z,Y,X,) or in any other sequence as long as the sequence is maintained throughout the calculations. Where alternate lattices are available for coordinate selection the lattice columns may be interchanged since this will effect the interchange of coordinate order and the results will be the same as long as the interchanged column order is maintained throughout the use or calculations). 
     It should be emphasized however that for the preferred detection method and means taught herein the modified Leech matrix or a matrix with the same column arrangements must be used at the detector stage. 
     Since the Pattern Matrix provides 4096 pattern vectors and coordinates modified by each of these patterns may be even or odd, the number N which identifies the bit sequence in the block of b bits is divided by 8192 to provide a quotient M and a remainder C. The quotient M will be encoded to 24 coordinates of a message point on the Z lattice and, after such encoding, the coordinates will be modified in accord with the selected pattern vector and the choice of even or odd coordinates, both being determined by the number C. 
     In the encoding algorithm of the embodiment Z tables are provided for dimension 1, 2, 4, 6, 12 and 24. Each dimension table, called Z1, Z2, Z4, etc., tabulates for shell number the number of points which satisfy the coordinate rules (the coordinate modulo sum rules are not considered except in 24 dimension). The shell number in each table is 1/16 the square of the radius (r 2 ) measured from the origin, in the dimension being considered, to each of the set of coordinates making up the entry for that shell number. The value of 1/16 is derived from the fact that each coordinate value is separated from the next value by four units (in accord with the modulo rules for the coordinates as distinct from the rules for the coordinate sum) so that the coordinates of a point, in whatever dimension, will occur at some of the steps r 2  =0, 16, 32, 48 etc., from the origin. 
     Each Z table is constructed from the next in accord with the combining algorithm ##EQU3## (where H, F and G are dimensions and F+G=H) for unit steps of I. This is worth noting since the splitting algorithm used to determine the coordinates corresponding to a number M is based on the structure of the above algorithm used to make the tables. The Z24 table which is the basis for the `Offset Table` used with the encode, detect, decode algorithms, does not include the values for the odd shell numbers, which the above algorithm would provide, since such the coordinates of points on such odd numbered shells will not satisfy the 0 modulo 8 requirements of the coordinate sum for 24 dimensions (all even coordinates) and The Offset Table provides, for each even shell number, the total number of points on the previous shells in the sequence defined by the shell table. By locating the Offset Table entry with the highest value not exceeding M the shell is identified where the message point represented by M will be located and the value M less the Offset table value, gives a value D (otherwise referred to as X 24 ) which identifies the point on the selected shell. By use of the splitting algorithm forming part of the encoding algorithm the value D which identifies a point on the selected 24 dimensional shell is used to derive first number X 12 (F) on a first 12D shell and a second number X 12 (G) on a second 12D shell. Continued use of the splitting algorithm as described in the encoding algorithm and will eventually derive 24 selected one dimensional shell numbers and for each an accompanying value which will be 0 or 1. Each of the Z lattice coordinates may be derived from the one dimensional shell number by taking the square root of the shell number and multiplying by 4. (It will readily be appreciated that one dimensional coordinate values could have been used throughout instead of the shell number, thus avoiding the last computation. However it is, at least conceptually, easier to use shell numbers). The 24 derived Z lattice coordinates define a point on the Z lattice defining quotient M. The coordinates are then modified in accord with a pattern vector derived from the modified leech lattice to take into account the value C. Even or odd coordinates are selected in accord with whether C is even or odd, 12 higher binary places of C define a number from 0 to 4095 which is used to select one of the 4096 pattern vectors. The coordinates are then modified as follows: (1) If C is odd, subtract 3 from the first of the 24 lattice point coordinates if the first bit of the pattern vector is a zero or add 3 if it is a one. For the remaining 23 Z lattice point coordinates add 1 if the corresponding pattern bit is a zero or subtract one if it is a one. It will be obvious that any coordinate could have been varied by 3 instead of the first although using the first is easier for programming purposes. It will further be obvious that the roles of the ones and zeros in the pattern vector may be reversed. (2) If C is even add 2 to each Z point lattice point coordinate for which the corresponding pattern vector is a one. It will also be realized that the roles of the ones and zeros can be reversed with the even C pattern vector. It will of course be obvious that such alterations in the encode program will require corresponding changes in the decode program. 
     In the encode algorithm which follows &#34;S&#34; stands for &#34;shell&#34; number. In such algorithm two shell numbers are involved I is one of these and J is the other. In tables Z1, Z2, Z4, Z6, Z12, the number on the left is the shell number and the number on the right is the number of available points in that shell noting that in the Z lattice the coordinates are all 0 modulo 4. In the offset table the shell number is the left column and the number in the right column is the total number of available points in the shells preceding the row in question. The total number of available points is limited to those satisfying the coordinate requirements and the modulo requirements for the coordinate sum in the offset table. 
     The principle of the splitting algorithm, as used in steps 4-26 of the encoding algorithm may be demonstrated by FIG. 2. A point defined as number D on a shell S24 in 24 space may be also defined by number R12A on shell S12A in 12 dimensional space together with number R12G shell S12G in 12 dimensional space. Each number and shell in 12 dimensional space may be identified by a pair of number and shell combinations in 6 dimensional space, and so on. 
     It will thus be seen that when in accord with the encoding algorithm the entry in the offset table is located with the largest offset number less than M, then the shell on which the point corresponding to D (equivalent to X 24 ) has been found and the value D=M-the offset number identifies a point on the shell, and the combination of shell number and D identify the number M. 
     The splitting algorithm described is then used to sequentially replace each combination of a shell number and a number identifying a point on the shell, in a dimension H, with two combinations of shell number and a point identifying number in dimensions F and G where F+G=H. This process is continued with F=G=1 whereby the coordinates and their signs can be identified. The `tree` of operations of the splitting algorithm is indicated by downward travel in FIG. 2. 
     In the encoding algorithm to follow the following points are noted. The algorithm relates to signalling using 24 message point coordinates. Since there are 4096 lattice centres and each may be used with 24 coordinates with the choice of making them all even or all odd the number N identifying the bit sequence in a block of b bits is divided by 8192 producing a quotient M and a remainder or coset point C. The designation D (X 24 ) refers to the general discussion where `X K  ` designates a point on a shell in K space and the designation D is used for X 24  in the algorithm to follow. 
     Further in the encode, detection and decode algorithms used herein the symbol `*` represents the multiplication operation or `x`. 
     It will be noted, that in step 4 of the splitting algorithm `X` is divided &#34;by entry I of table A&#34;. It will be appreciated that what is happening here is that the number X is to be assigned to two shells is being divided by entry I representing the first assigned shell&#39;s capacity of available points, to obtain a quotient Q to be assigned to the second assigned shell and a remainder (R), to be assigned to the first assigned shell. Obviously the roles can be reversed and division may be performed by the entry for second shell to obtain a quotient to be assigned to the first and a remainder to the second. This is equivalent to the procedure outlined if a corresponding complementary operation is performed at the decoder. 
     Also it will be noted that the sequence of shells represented by the offset table need not be in the order of rising shell number although this appears the most convenient for programming. However, the tables Z1-Z12 should be in order of rising shell number for convenience and simplicity in programming. 
     The choice of even or odd coordinates, together with the selection of pattern vectors all having determined the coordinates C1, C2, . . . C24 by the encoding algorithm these are provided by the microprocessor encoder 20 to the coordinate signal generator 30 which generates modulating signals in accord with the coordinate values. The modulating signals from coordinate generator are modulated on the carrier at modulator 45, to provide the modulated carrier signal to the transmitter channel interface 50. The type of modulation used is not limited by the invention and may be any of a large number of types as previously discussed. Most commonly QAM (quadrature amplitude modulation) will be used. 
     The signal transmitted from the transmitter channel interface 50 on the channel is received at the receiver channel interface 60. The signals received are subjected to conditioning, demodulation and equalization at block 70, all blocks 15,10,30,40,45,50,60,70,90 and 100 and being designed and operated in accord with techniques well known to those skilled in the art. 
     The conditioned, demodulated and equalized signals having the values C1-1/2, C2-2 . . . C24-1/2 are supplied to the detector 80 shown in FIG. 2 and described in the programming instructions for the detection algorithm described hereafter. The following comments on the detection algorithm programming steps, should be read therewith. 
     In accord with the detection method of the invention the transmitted coordinates which have been encoded and modulated at the transmitter are, at the receiver, demodulated. The demodulated received values have 1/2 added to compensate for the deduction in step 32 of the encoding algorithm. The received values each augmented by 1/2 are stored. Also stored are their values mod 4 (detection algorithm step 2), and mod 8. 
     Some aspects of the following discussion and the detection algorithm depend upon the consideration of any set of 24 coordinates in groups of 8, that is group 1 (g 1)-coordinates 1st to 8th; group 2 (g 2)-coordinates 9th to 16th and group 3 (g 3)-coordinates 17th to 24th. Such groups are known as first, second and third octuples herein. 
     It is possible to demonstrate that within the 8192 possible vectors (4096 pattern vector, multiplied by 2 for the choice of even or odd coordinates) that there are 256 possible values for the coordinates (mod 4) in an octuple which may be arranged in 128 complementary pairs. (See these complementary pairs in Table 2). 
     It is possible to demonstrate that of the possible pattern vectors there are only 1024 patterns of the first and second octuples (taken together and sometimes referred to as a dual octuple) which may be arranged in 128 groups of eight. 
     The demonstration is: The detection algorithm is based on the following derivation from the conventional Leech matrix. It may be noted that the Leech matrix becomes an equivalent matrix and produces the Z and coset lattices with any different sequence of coordinates and furthermore any rows may be replaced by the addition of mod 2 of other rows as long as all rows are represented as components of sums in the final matrix to avoid a singular matrix. The following form may be obtained from the conventional Leech matrix by re-arranging the columns into the sequence: (numbers are of conventional matrix columns) 1,18,24,22,19,10,23,3/6,8,21,16,12,15,17,7,/9,13,2,4,12,5,20, 11. Th strikes indicate the division between octuples. 
     And then by creating new rows from the sums (mod 2) of the following old rows: 
     
         ______________________________________                                    
New Row          Old Row Abbreviated                                      
______________________________________                                    
1                1,3,10                                                   
2                6,7,8,12                                                 
3                2,4,5,9,11                                               
4                3,7,10,12                                                
5                3,7                                                      
6                3,7,8,10                                                 
7                5,7,11,12                                                
8                2,4,7,11                                                 
9                4,5,7,8,11                                               
10               3,5,7,8,10,11,12                                         
11               3,4,7,8,11                                               
12               2,4,5,8,10                                               
______________________________________                                    
 
    
     See Row Addition Table for full list. 
     Thus producing: 
     
         ______________________________________                                    
1       11111111      00000000 00000000                                   
2       00000000      11111111 00000000                                   
3       00000000      00000000 11111111                                   
4       00001111      00001111 00000000                                   
5       00110011      00110011 00000000                                   
6       01010101      01010101 00000000                                   
7       00000000      00001111 00001111                                   
8       00000000      00110011 00110011                                   
9       00000000      01010101 01010101                                   
10      01001101      01001101 01001101                                   
11      01010011      01010011 01010011                                   
12      01110100      01110100 01110100                                   
______________________________________                                    
 
    
     This is referred to herein as the `Modified Leech Matrix`. 
     The essential features of this form are: that the first three rows have all zeros in two of the three octuples and all ones in the other octuple (in successively the first, second and third octuples). the next three rows have two identical octuples followed by eight zeros and the next three rows have eight zeros followed by two of these same 8-tuples. The final three rows each consist of three identical 8-tuples. 
     Any subset of rows of the above matrix may be selected and exclusive ored together (i.e. summed modulo 2) to produce one of 4096 possible pattern vectors. 
     The coordinate rules for 24 place coordinate sets derived from the Leech lattice are then: 
     They may be 24 even integers which are 0 mod 4 where a pattern vector has a 0 and 2 mod 4 where the same pattern vector has a 1 and the sum of the coordinates is 0 mod 8. 
     Or they may 24 old integers which 1 mod 4 where a pattern has a 0 and 3 mod 4 where the pattern vector has a 1 and whose sum is 4 mod 8. 
     The &#34;detection problem&#34; is: Given an arbitrary point in 24-space as represented by 24 given coordinates, determine the pattern vector used and the choice of odd or even coordinates associated with the Z or co-set lattice point closest to the given point. 
     The essential feature of this detection method may be seen by noting: 
     Within any octuple there are only 128 different patterns and that these fall into 64 pairs which differ by 11111111 (and therefore are complements since the rows are added, mod 2) as determined by rows 1, 2 or 3. 
     Within the first 16 coordinates (or dual octuples), rows 4 to 12 produce only 512 patterns which may be divided into 64 groups of 8 where the members of a group differ only by the choice of rows 4, 5 and 6. 
     It has previously been explained that columns in the Leech matrix used, may be interchanged as long as the same conversion is used at transmitter and receiver as performed in the preferred embodiment, or in an alternate method in accord with the invention at the beginning and end of the pattern vector detection process. (In the alternate method it will be noted that one version of the matrix can be used during encoding, modulating transmission and initial reception and demodulation. After demodulation the places of the coordinates may be interchanged to render detection easier. This has, at this stage, (since the characteristics of a pattern vector are impressed upon the coordinate values), the same effect as the interchanging of matrix columns before the pattern vector is selected. After the detection of the pattern vector used, the coordinates and the pattern vector values may have their order returned to that transmitted and originally received, for further processing.) In the preferred method the 24 coordinates representing a block encoded group of bits have been chosen at the transmitter (a) to satisfy the individual coordinate modulo rules and the coordinate sum modulo rules and (b) for modification in accord with a particular Leech pattern vector; and (c) whether they are to be all even or all odd. Such modified coordinates are modulated on a carrier and transmitted. Such modulation, whatever they type used, is usually in accord with a quadrature system and under such system the coordinates are modulated on the carrier in pairs. 
     At the receiver the detected signals are demodulated and the received values of the coordinates stored. Such received coordinates differ from those transmitted due to the effects of distortion, noise, and other causes during modulation, encoding and transmission and the problem of detection is, of course, to determine the pattern vector used and the actual coordinates encoded. 
     In the detection algorithm: 
     For detection of the pattern vector used the process uses C n  the value of the demodulated coordinates where n=1 to 24, is the number of the coordinate. X n  is C n  mod 4. 
     X n  is composed of I n , an integral and F n , a fractional part. It will be noted that, (since X n  is calculated mod 4) the value I n , may be 0, 1, 2, 3. k is the possible transmitted value of I n  and likewise varies through 0, 1, 2, 3. 
     A table (Table 1) is provided having, for each value I n , the four values of k. For each of such combinations there is calculated: 
     M n ,k : a measure of the square of the distance from X n  to k, a `measure of the error` 
     P n ,k : a positive measure of the difference of squared distance M n ,k and the squared distance from X n  to the closer of k+4, k-4. Thus P n ,k represents the penalty or increase in the value M n ,k if the closer of k+4 or k-4 has later to be substituted for k to achieve the modulo sum rules for 24 coordinates. 
     B n ,k : is an 0 or 1 value designed to give, for each of the 24 coordinates, an indication of the effect of the coordinate on the modulo sum requirements. If I n  mod 4 does not equal the integral value of C n  mod 8 then the value for B n  in Table 1 must be complemented. 
     In a particular combination of circumstances, as described hereafter, where complementary octuples are being compared &amp; B n ,k is different for the two; P n ,k will sometimes be increased to cover the increase of error measure on switching from one to the other octuple. 
     Table 2 contains 128 pairs k1,k2 of octuple groups (j) of coordinates modulo 4. In each pair the two octuples are complements, that is, for even pairs, the 2&#39;s and 0&#39;s are interchanged and, for odd pairs the 1&#39;s and 3&#39;s are interchanged. 
     (It may be noted that such pairs occur due to the effect of the first three rows of the modified Leech matrix. Thus the production of a complement value for the first second, third, octuple g1,g2,g3, is due to the fact that each place in the octuple contains a one if the first, second, third, row respectively, is included and zero if the others of rows 1, 2 or 3 are included. Thus from the selection of k1 vs k2 values in the first 24 mod 4 coordinates, 3 of the 13 bits are determined indicating the combination of modified matrix rows one to three used). 
     For each pair of Table 2 the group sums, SUM of the values M n ,k over an octuple are compared and the octuple of the pair with the lower SUM is selected GM g ,j. A record of which octuple is selected is stored. RN g ,j =0,1 corresponds to SUM for k1 smaller, larger than k2 respectively. Records are made of the storage and use of the penalty values P1 in case a switch or `flip` must be made from the closest to the next closest k+4 value to satisfy the modulo sum rules for 24 coordinates. 
     Penalty values must be assembled in case of the necessity for a `flip`. The term `flip` refers to the situation where the 24 mod 4 coordinate total does not satisfy the coordinate sum modulo 8 rule, and it is necessary to find the change of k to k+4 or k-4 or alternate change that will cause the smallest increase in GH g ,j such increase being the smallest penalty P n ,k. 
     The general approach to calculating the penalty GP g ,j for a given octuple from table 2, reflected in paragraphs 3 is to note the smallest P n ,k for each octuple. 
     The sum SUM 1 of the M n ,k1 values is compared with the sum SUM 2 of the M n ,k2 values and the smaller designated GM g ,j. 
     1. If the contribution of B nj  to the mod 8 sum for 24 coordinates is the same for both k1 and k2 values of an octuple pair then the `flip` value SUM 1+ smallest P n ,k1 is compared with SUM 2+ smallest P n ,k2. 
     (a) If SUM 1&lt;SUM 2 and the flip values involving SUM 1&lt; those involving SUM 2 then SUM 1=GM g ,j and P n ,k1 =GP g ,j. 
     If both inequalities are reversed the SUM 2=GM g ,j and P n ,k2 =GP g ,j. 
     (b) If SUM 1&lt;SUM 2 but SUM 1+ smallest P n ,k1 is &gt; than SUM 2+ smallest P n ,k2 then SUM 1=GM g ,j but the penalty GP g ,j is SUM 2+ smallest P n ,k1 -SUM 1. If both inequality signs are reversed then SUM 2=GM g ,j and GP g ,j is the SUM 1+ smallest P n ,k1 -SUM 2. 
     Thus what has been determined here is, GM g ,j --a measure of the error value for the lower table of a pair, and also the penalty or increase in error measure is the modulo 8 increment of the octuple must be altered, and whether a lower penalty is incurred by a `flip` of one coordinate of the octuple or by switching to the other octuple and flipping a coordinate of the latter. 
     2. If the combination of B n ,j to the mod 8 sum for 24 coordinates is different for the k1 and k2 values of an octuple pair then if a flip is required for the smaller SUM=GM g ,j the penalty by flipping a coordinate of the selected octuple is compared with the penalty by substituting the other octuple and the lower penalty stored as GP g ,j with a stored direction whether, if such flip is required, the coordinate is flipped or the other octuple substitute. 
     The G m ,g values and RN g ,1 the 128 pairs winners are identified with the corresponding &#34;j&#34; value of Table 2. It will be noted that for any of the three possible octuples (mod 4) making up the 24 coordinates, mod one of two choices has been eliminated and this result is the equivalent of 3 bits of information of the 13 required, that is the 12 bits to identify the modified matrix rows use to make the pattern vector and one to identify the selection of whether even or odd coordinates have been chosen. 
     Table 3 values are expressed in base 8 (octal) values for brevity. Thus each octal digit corresponds to three consecutive binary digits. 
     Comprising table 3 with the modified Leech matrix it will be noted that, in each row of eight dual octuples, the only difference between such dual octuples is the eight possible combinations of use of rows 4, 5 and 6 of the modified Leech matrix. It may also be noted that the same j3 value may be used for each of the J1,j2 dual octuple pairs because the third octuple of each of rows 4, 5 and 6 of the modified matrix consists of eight 0&#39;s. 
     As elsewhere described the summing of the values GM n ,j for the first two octuples as paired by table 3 are summed to determine lowest GM n ,1 +GM n ,2 for each of the 128 sets of eight. 
     It should be noted that the lowest sum for each group of eight may be determined by summing corresponding to only the first two octuples j1,j2. Adding the third octuple is unnecessary because, for any row group of eight j3 is the same for each member of the group. If, instead of rows 4, 5 and 6, the rows 10, 11 and 12 had been used to determine the components of the eight row sets of table 3, j3 would have been different for each row set and the winner of the row set would have had to be determined by summing j1, j2, j3 for each row in the set. On the other hand if rows 7, 8, 9 had been used to distinguish the eight components of the row sets (because of the eight zeros in each first octuple) j1 would have been the same for the J2 and j3 values would have been set out as the dual octuples of table 2. 
     With the 128 candidates from the best candidates of the row sets of Table 3 then to each sum GM n ,1 +GM n ,2 is added the GM n ,3 as determined by the Gm n ,j3 of the j3 of the table. The B n ,1 +B n ,2 +B n ,3 values are similarly summed (mod 2). If the modulo sum, as indicated by the summed B values, is incorrect then the coordinate lowest GP n ,j is added with or without the substitution of a complementary octuple as previously described. 
     The 128 error measure sums ##EQU4## sometimes augmented by GP n ,j are used to determine the winner and if appropriate to substitute a complementary octuple as previous discussed. Thus the lowest error measure sum of the 128 candidates is the winner. 
     With such `winner of 128` the 13 bits to identify even and odd and the Leech matrix pattern vector can be determined, as discussed and, as shown are indicated by the octal digits in the `bits` column of Table 3. 
     The GM g ,j, GP g ,j, GB g ,j, RN g ,j, RF g ,j for the 128 pair winners are identified with the corresponding &#34;j&#34; value of Table 2. It will be noted that for any of the three possible octuples (mod 4) making up the 24 coordinates, one of two choices has been eliminated in step 2 and this result is the equivalent of 3 bits of information of the 13 required, (that is the twelve bits to identify the pattern vector and one to identify whether even or odd coordinates have been chosen). 
     Since row k 2  is the obvious complement of k 1  it will be noted that k 2  will have a lower G m  when rows 1, 2, 3 of the modified Leech matrix have been used for (what turns out to be) the first octuple, second octuple and third octuple respectively, since the eight 1&#39;s in the matrix octuple will (mod 2 addition) produce the complement. Thus these k 1  v k 2  candidates in the three octuples in the final result will provide the first three bits of the row determinant. 
     The `row determinant` is the series of 12 binary digits which determines which of the rows of the Leech or modified Leech matrix are summed, mod 2 to produce the pattern vector. Thus the row determinant 010 010 010 010 states that the second, fifth, eighth and eleventh rows of the modified Leech matrix should be added mod 2 to produce the pattern vector. 
     Table 3 gives the 512 possible combinations of the first two octuples j1, j2 in pattern vectors derived from a modified Leech matrix. Actually Table 3 gives 256 values which are doubled by considering there is a further table differing from the first only in that a 1 is added before each j1, j2 (and j3 in a later step). 
     To be consistent with the previous discussion, and in use, j1, j2, j3 are binary digits. However for compactness herein these j values are expressed in octal notation and each octal digit represents three successive binary digits. 
     In Table 3, the eight sets of j1, j2 values in each row convey through the first octal digit in the bits column, the presence of the modified matrix rows (reading the rows from left to right): 
     
         __________________________________________________________________________
Octal Digits bits column                                                  
            0XX                                                           
               1XX                                                        
                  2XX                                                     
                     3XX 4XX                                              
                            5XX 6XX 7XX                                   
__________________________________________________________________________
Rows 4,5 or 6 of                                                          
            0* 4  5  4 &amp; 5                                                
                         6  6 &amp; 4                                         
                                6 &amp; 5                                     
                                    654                                   
modified matrix                                                           
Binary Digits**                                                           
            000                                                           
               001                                                        
                  010                                                     
                     011 100                                              
                            101 110 111                                   
for row determinant                                                       
__________________________________________________________________________
 *only rows 4,5 and 6 are excluded other rows may be included.            
 **each of these will be followed by six bits which may be 0 or 1 as      
 otherwise determined and preceded by 3 bits determined by RN.sub.1,      
 RN.sub.2, RN.sub.3.                                                      
 
    
     Thus the presence of rows 4, 5 and 6 in the mod 2 sum is determined by the three bits which are set out above and the order would normally be reversed to place the bits indicating use of rows 4,5 and 6 in places 4,5 and 6 respectively, of the row determinant. 
     With each of the 128 winners of the dual octuple sets, including the 64 starting with the added 1 to j1 and j2, and j3 value is added with a preceding 0 or 1 to agree with the j1 and j2 values. 
     The values ##EQU5## are added for the j1, j2, j3 sometimes augmented by a GP n ,j in case of each of the 128 dual octuple winners and the lowest sum determines the winner. The effect of the determination of places 4,5 and 6 of the row determinant has already been described. In the &#34;best of 128&#34;  competition and Table 3, it will be noted that for each set of eight rows (e.g. bits XX0 to XX7) the last digit determines the inclusion or not of rows 7,8 and 9 of the modified matrix in the pattern vector modulo 2 sum as follows: 
     
         ______________________________________                                    
Octal Digits            Rows 7,8 or 9                                     
bits Column  Binary Digits                                                
                        of modified matrix                                
______________________________________                                    
XX0           000**      0*                                               
XX1          001        7                                                 
XX2          010        8                                                 
XX3          011        7 &amp; 8                                             
XX4          100        9                                                 
XX5          101        9 &amp; 7                                             
XX6          110        9 &amp; 8                                             
XX7          111        9,8 &amp; 7                                           
______________________________________                                    
 *none of rows 7,8 or 9 but may have some of rows 1-6 or 10-12.           
 **preceded by 9 binary digits.                                           
 
    
     Thus the seventh to ninth places of the row determinant may be chosen to indicate the presence or absence of rows 7,8 or 9 in the modulo 2 sum. 
     In the &#34;best of 128&#34; competition and Table 3, it will be noted that for each of the eight sets of sets of eight rows (e.g. with bits values X0X to X7X, the middle octal digit represents the extent of usage of rows 10,11,12 of the modified matrix. The middle octal digit determines the inclusion of rows 10,11 and 12 as follows: 
     
         ______________________________________                                    
Octal Digits*           Rows 10,11,12 of                                  
bits Column Binary Digits**                                               
                        modified Leech matrix                             
______________________________________                                    
X0X         000           0***                                            
X1X         011         10                                                
X2X         010         11                                                
X3X         011         11 &amp; 10                                           
X4X         100         12                                                
X5X         101         12 &amp; 10                                           
X6X         110         12 &amp; 11                                           
X7X         111         12,11 &amp; 10                                        
______________________________________                                    
 *chosen from successive vertical sets of eight.                          
 **preceded by six and succeeded by three bits.                           
 ***none of rows 10,11 or 12 but may have some of rows 1-9.               
 
    
     Thus the tenth to twelfth places of the row determinant may be chosen to indicate the presence or absence of rows 10,11 or 12 in the modulo 2 sum. 
     Thus the twelve digits indicating the rows combined from the modified Leach matrix to produce the pattern vector may be ascertained. 
     The first number of any j that is the added 1 or 0 of &#34;the best of 128&#34; winner will determine even or odd. 
     Thus the bits in the row determinant and hence (if the bit is one) the use of the corresponding rows in the mod 2 sums are determined as follows (left to right): 
     Row determinant places: 
     1, 2, 3 
     from the k 1  v k 2  &#34;winners&#34; determined in accord with Table 2, by such winners which were determined to be octuple members j1, j2, j3 of the winner. Presence of a 1 in a place of the row determinant corresponds to use of the row from the modified matrix in the mod 2 sum to determine the pattern vector. 
     4,5,6 
     by the column in table 3, containing the &#34;winner of 128&#34;, the presence of absence of rows 4,5 and 6 in the modified matrix and corresponding places in the row determinant being represented by the binary digits corresponding to the first octal number in the bits column in the order 6,5,4. 
     7,8,9 
     by the choice of the row (mod 8) in the &#34;best of 128&#34; choice. The presence or absence of rows 7,8 or 9 in the modified matrix and corresponding places in the row determinant being represented by the binary digits corresponding to the third octal digit of the bits column in the order 9,8,7. 
     10,11,12 
     by the second octal digit in the bits column of the &#34;best in 128&#34; choice representing which set of sets of eight rows which octal digit when written in a binary form represents the presence or absence of rows 10,11,12 in the modified matrix and corresponding places in the row determinant being represented by the binary digits corresponding to the second octal digit of the bits column in the order 12,11,10. 
     Even or Odd 
     by the choice of 0 or 1 at the beginning of the &#34;best of 128&#34; competition. 
     When the detection algorithm is set out hereafter, subscripts are not used so that M n ,k, P n ,k and B n ,k etc. become M(n,k) P(n,k) and B(n,k) etc. 
     The determination of the pattern vector corresponds to step 5 of the detect algorithm. As steps 6-8, the row determinant represented by the first twelve places of C is used to determine the pattern vector which is used (steps 7 or 8) to modify the detected coordinates (full not mod values) to reverse the modification in accord with the same pattern vector at the encoder. Step 9 alter stored coordinate value corresponding to a flipped coordinate. 
     The discussion to follow corresponds to the decoding algorithm to follow. 
     In the decoding algorithm the 1D Z lattice points S1(1), R1(1); S1(2) R1(2) . . . S1(24) R1(24) are combined by the combining algorithms to determine the encoded number Z. The combining algorithm reverses each of the steps of the splitting algorithm. The combining algorithm is further used to reverse the dimensional steps represented by the tree of FIG. 2 to produce the value D,S24 from the 24 S1, R1 values. 
     With regard to the combining algorithm, it will be noted that in the splitting algorithm (dimension H=dimensions F+G) entries for ##EQU6## (n is the shell of H while I and J are the entries for the corresponding shell no), were performed with successive subtractions from X H  until I k  ×J n-k  &gt;last X. The entry for I k  was then divided into the last X to produce quotient Q and remainder R. 
     In the exact converse, in the combining algorithm, I k  is multiplied by the quotient Q and R is added to produce the &#34;last X&#34; of the splitting algorithm. The first X of the splitting algorithm is provided from the last X in the combination algorithm by adding the products of the entries for I k  -i×J n-k+1  until k-i=0. The result of the combining algorithm is to produce the number X H  and the shell number in dimension H having started with two pairs of values X F  and shell no in dimension F and, X G  and shell no in dimension G. 
     It will readily be appreciated that at step 25 of the decoding algorithm the shell number gives the offset value for addition to D, to give the encoded sum Z. Z is multiplied by 8192 reversing the division at the encoder. When C is to be added, it will be noted that the value C was obtained by the micrprocessor 90 from register 15. 
     The value N is then transmitted as b bits from microprocessor 90 to parallel to series convertor 100 were it is converted to serial binary bits. Assuming that scrambling was performed at scrambler 15 than the serial binary bits are unscrambled at uncrambler 95 into unscrambled serial binary bits of serial binary data. 
     In considering the encoding, detection and decoding algorithms the following points should be noted. 
     Although the algorithms give only one tabulation of the pattern matrix the offset table and tables Z1-Z12; it will be obvious that such tables exist both in the microprocessor 20 at the transmitter end of the channel for use with the encoding algorithm and in the microprocessor 90 at the receiver end of the channel for use with decoding algorithm. 
     X 24  identifies a message point or a shell in 24 dimensions. X 24  corresponds to D in step 3 of the encoding algorithm and step 24 of the decoding algorithm. 
     The encoding, detection and decoding algorithms are set out below: 
     ENCODING ALGORITHM 
     At a data rate of 19,200 bits per second, the data to be transmitted in one frame consists of 96 bits which may be considered as a number, N, in the range from 0 to 79,228,162,514,264,337,593,543,950,335. If an auxillary 200 bps channel is desired, the number of bits is increased to 97 so that N is from 0 to 158,456,325,028,528,675,187,087,900,671. At 16,800 bits per second with an auxillary channel, only 85 bits are required which may be considered as a number, N, in the range from 0 to 38,685,626,227,668,133,590,597,631. Similarly, at 14,400 bits per second, only 73 bits are required which may be considered as a number, N, in the range from 0 to 9,444,732,965,739,290,427,391. The number, N, may be converted to a group of 24 coordinates to be transmitted by the following algorithm. 
     Step 1: Divide N by 8192. Let the quotient be M and the remainder be C. M will now be encoded into a point in the Z lattice and C will be used to select a coset point. 
     Step 2: Scan the offset table (table 1) to find the entry for which the value in the offset column is as large as possible but does not exceed M. Let S be the corresponding shell number. 
     Step 3: Subtract the value in the offset column from Z giving a difference D. So far we have selected a shell corresponding to the table entry and we will use the value of D to select a message point from within this shell. 
     Step 4: Set X=D and J=S. Then use the splitting algorithm (described later) with Table A=table Z12 and Table B=table Z12. Let S12A=I, S12G=J, R12A=R and R12G=Q 
     Step 5: Set X=R12A and J=S12A. Then use the splitting algorithm with Table A=table Z6 and Table B=table Z6.  Let S6A=I, S6D=J, R6A=R and R6D=Q 
     Step 6: Set X=R12G and J=S12G. Then use the splitting algorithm with Table A=table Z6 and Table B=table Z6. Let S6G=I, S6K=J, R6G=R and R6K=Q 
     Step 7: Set X=R6A and J=S6A. Then use the splitting algorithm with Table A=table Z2 and Table B=table Z4. Let S2A=I, S4B=J, R2A=R and R4B=Q 
     Step 8: Set X=R4B and J=S4B. Then use the splitting algorithm with Table A=table Z2 and Table B=table Z2. Let S2B=I, S2C=J, R2B=R and R2C=Q 
     Step 9: Set X=R6D and J=S6D. Then use the splitting algorithm with Table A=table Z2 and Table B=table Z4.  Let S2D=I, S4E=J, R2D=R and R4E=Q 
     Step 10: Set X=R4E and J=S4E. Then use the splitting algorithm with Table A=table Z2 and Table B=table Z2. Let S2E=I, S2F=J, R2E=R and R2F=Q 
     Step 11: Set X=R6G and J=S6G. Then use the splitting algorithm with Table A=table Z2 and Table B=table Z4. Let S2G=I, S4H=J, R2G=R and R4H=Q 
     Step 12: Set X=R4H and J=S4H. Then use the splitting algorithm with Table A=table Z2 and Table B=table Z2. Let S2H=I, S2J=J, R2H=R and R2J=Q 
     Step 13: Set X=R6K and J=S6K. Then use the splitting algorithm with Table A=table Z2 and Table B=table Z4. Let S2K=I, S4L=J, R2K=R and R4L=Q 
     Step 14: Set X=R4L and J=S4L. Then use the splitting algorithm with Table A=table Z2 and Table B=table Z2. Let S2L=I, S2M=J, R2L=R and R2M=Q 
     Step 15: Set X=R2A and J=S2A. Then use the splitting algorithm with Table A=table Z1 and Table B=table Z1. Let S1(1)=I, S1(2)=J, R1(1)=R and R1(2)=Q 
     Step 16: Set X=R2B and J=S2B. Then use the splitting algorithm with Table A=table Z1 and Table B=table Z1. Let S1(3)=I, S1(4)=J, R1(3)=R and R1(4)=Q 
     Step 17: Set X=R2C and J=S2C. Then use the splitting algorithm with Table A=table Z1 and Table B=table Z1. Let S1(5)=I, S1(6)=J, R1(5)=R and R1(6)=Q 
     Step 18: Set X=R2D and J=S2D. Then use the splitting algorithm with Table A=table Z1 and Table B=table Z1. Let S1(7)=I, S1(8)=J, R1(7)=R and R1(8)=Q 
     Step 19: Set X=R2E and J=S2E. Then use the splitting algorithm with Table A=table Z1 and Table B=table Z1. Let S1(9)=I, S1(10)=J, R1(9)=R and R1(10)=Q 
     Step 20: Set X=R2F and J=S2F. Then use the splitting algorithm with Table A=table Z1 and Table B=table Z1.  Let S1(11)=I, S1(12)=J, R1(11)=R and R1(12)=Q 
     Step 21: Set X=R2G and J=S2G. Then use the splitting algorithm with Table A=table Z1 and Table B=table Z1. Let S1(13)=I, S1(14)=J, R1(13)=R and R1(14)=Q 
     Step 22: Set X=R2H and J=S2H. Then use the splitting algorithm with Table A=table Z1 and Table B=table Z1. Let S1(15)=I, S1(16)=J, R1(15)=R and R1(16)=Q 
     Step 23: Set X=R2J and J=S2J. Then use the splitting algorithm with Table A=table Z1 and Table B=table Z1. Let S1(17)=I, S1(18)=J, R1(17)=R and R1(18)=Q 
     Step 24: Set X=R2K and J=S2K. Then use the splitting algorithm with Table A=table Z1 and Table B=table Z1. Let S1(19)=I, S1(20)=J, R1(19)=R and R1(20)=Q 
     Step 25: Set X=R2L and J=S2L. Then use the splitting algorithm with Table A=table Z1 and Table B=table Z1. Let S1(21)=I, S1(22)=J, R1(21)=R and R1(22)=Q 
     Step 26: Set X=R2M and J=S2M. Then use the splitting algorithm with Table A=table Z1 and Table B=table Z1. Let S1(23)=I, S1(24)=J, R1(23)=R and R1(24)=Q 
     Step 27: Generate the 24 coordinates of the Z lattice point from S1(1) to S1(24) and R1(1) to R(1)24. Each coordinate is 4 times the square root of the corresponding S1(N). If R(1)n=1, the sign of the coordinate is to be made negative. 
     Step 28: The 12 high order bits of the 13 bit binary representation of C are associated with the twelve rows of the pattern matrix with the most significant bit associated with the first row. The 12 high order bits of C constitute the row determinant. 
     Step 29: Exclusive or together those rows of the matrix for which the associated bit of C is a one to produce a &#34;pattern vector&#34; of 24 bits. 
     Step 30: If C is even, add 2 to each Z lattice point coordinate for which the corresponding pattern vector bit is a one. 
     Step 31: If C is odd, subtract 3 from the first Z lattice point coordinate if the first bit of the pattern vector is a zero or add 3 if it is a one. For the remaining 23 Z lattice point coordinates, add 1 if the corresponding pattern bit is a zero or subtract one if it is a one. 
     (Coordinates C1, C2 . . . C24 have now been produced with step 30 or 31.) 
     Step 32: Subtract 1/2 from each coordinate to remove the statistical bias introduced in step 30. The coordinates are now ready for transmission. 
     The splitting algorithm invoked above is as follows: 
     Step 1: Set I=0. 
     Step 2: Multiply entry I of table A by entry J of table B to produce a product, P. 
     Step 3: If P is less than or equal to X, Subtract P from X (the difference is a new X). Add 1 to I, Subtract 1 from J and return to step 2. If P is greater than X, continue to step 4. 
     Step 4: Divide last X by last entry I of table A to produce a quotient, Q and a remainder R. The splitting algorithm is now complete. 
     DETECTION ALGORITHM 
     The detection algorithm proceeds as follows: 
     Step 1: Add 1/2 to each coordinate to compensate for the subtraction performed in the transmitter. 
     Step 2: For each of the 24 coordinates, C(n), (n=1 to 24) compute M(n,k), P(n,k) and B(n,k) (k=0 to 3) as specified by the table 1 where X(n) is the coordinate, C(n), modulo 4, I(n) is the integer part of X(n) and F(n) is its fractional part. If C(n) mod 8 is not equal to X(n), the values of B(n,k) must be complemented. For computational convenience, the single bit value of B(n,k) can be kept in the low order bit of M(n,k) thus automatically performing the sums modulo 2 whenever the corresponding Ms are added. Carries out of the B bit slightly contaminate the M values but provided that sufficient precision is used this is unimportant. 
     Note that two alternative formulae are given for M(n,k). The second is equal to the first+1-F(n)**2 and avoids the squaring. Either may be used throughout. 
     Step 3: Now divide the coordinates into 3 groups of 8 (g=1 to 3) where 1 to 8 form group 1, 9 to 16 form group 2 and 17 to 24 form group 3. 
     Compute GM(g,j), GP(g,j) GB(g,j) RN(g,j) RF(g,j) for the 128 values of j (which for convenience are numbered in octal notation from 000 to 177) and for each group of coordinates g in accordance with table 2 as follows: 
     For each j, the 8 digits from column k1 or k2 of table 2 define values of k for the eight coordinates. Compute temporary variable SUM1 as the sum of the selected values of M(n,k), PEN1 as the smallest of the selected values of P(n,k) and BIT1 as the sum modulo 2 of the selected values of B(n,j) using column k1 to select the values of k. Compute SUM2, PEN2 and BIT2 in the same way using column k2. 
     If SUM1 is less than SUM2, set GMg, j=SUM1, set GB(g,j)=BIT1 and set RN(g,j)=0; otherwise set GM(g,j)=SUM2, set GB(g,j)=BIT2 and set RN(g,j)=1. 
     Now set PEN1=PEN1+SUM1-GM(g,j) and PEN2=PEN2+SUM2-GM(g,j) (Note the one of SUM1-GM(g,j) or SUM2-GM(g,j) must be zero) 
     If PEN1 is less than PEN2, set GP(g,j)=PEN1 and set RF(g,j)=0 otherwise set GP(g,j)=PEN2 and set RF(g,j)=1. 
     If BIT1 is the same as BIT2, The rest of this paragraph is ignored. Otherwise compute TEMP=the absolute value of the difference between SUM1 and SUM2 and compare this with GP(g,j) as computed above. If TEMP is smaller, replace GP(g,j) with TEMP and replace RF(g,j) with RN(g,j)+1 (modulo 2). Any GP(g,j) replaced in this way should be marked as &#34;special&#34; 
     Note that the number of actual computation steps may be reduced at the expense of some complexity by first computing values for each of four coordinate pairs, combining the into values for each of two coordinate 4-tuples and finally computing the values required. 
     Note also that the computation of RN and RF may be omitted at this stage and computed only after the final winner is known at the end of step 4. 
     Step 4: Each row of table 3 contains a value of j3 followed by eight entries each giving a value of j1, j2 and a 9 bit octal value headed &#34;bits&#34;. Consider table 3 to be doubled in size with a leading 0 appended to each of the values of j in the first (even) half and with a leading 1 appended to each of the values of j in the second (odd) half. 
     For each of the 8 entries within a row sum GM(1,j1) and GM(2,j2). If the sum modulo 2 of GB(1,j1), GB(2,j2) and GB(3,j3) is 1 for an even row or 0 for an odd row add the smallest of GP(1,j1), GP(2,j2) and GP(3,j3) to the above sum. Select the smallest of the 8 sums and add GM(3,j3) to get the &#34;winner&#34; of this row. 
     The smallest of the 128 row winners identifies the final result. If the winning sum did not include a GP(g,j) term the output bits are RN(1,J1), RN(2,j2), RN(3,j3) followed by the nine bits from the &#34;bits&#34; column of the selected entry of the winning row followed by a 0 if the winner was even or a 1 if odd. If a GP(g,j) term was included in the winner, one of RF(1,j1), RF(2,j2) or RF(3,j3) must replace one of RN(1,j1), RN(2,j2) or RN(3,j3) depending on which of GP(1,j1), GP(2,j2) and GP(3,j3) was smallest. Of these 13 bits, the first 12 define the matrix rows used and the last defines odd/even. These 13 bits comprise the number C. 
     If a GP(g,j) term was included in the final winner, and it was not marked as &#34;special&#34; (see step 3), set FLIP=the number of the coordinate which contributed the smallest penalty, P. Otherwise set FLIP=0. 
     If the first Mn,k column of table 1 was used. The smallest sum is the square of the distance to the nearest lattice point. If the second column was used it may be corrected if desired by adding the sum of the squares of the fractional parts of the coordinates and subtracting 24. 
     Note that it is not necessary to completely perform one step before commencing the next. Each group of eight entries of table 2 provide sufficient data to perform step 4 with 8 rows of table 3. 
     Step 5: The 12 high order bits of the 13 bit binary representation of C are associated with the twelve rows of the pattern matrix with the most significant bit associated with the first row. These 12 high order bits constitute the row determinant. 
     Step 6: Exclusive or together those rows of the matrix for which the associated bit of C is a one to produce a &#34;pattern vector&#34; of 24 bits. 
     Step 7: If C is even, subtract 2 from each detected coordinate C1, C2 . . . C24 for which the corresponding pattern vector bit is a one. 
     Step 8: If C is odd, add 3 to the first detected coordinate C1 if the first bit of the pattern vector is a zero or subtract 3 if it is a one. For the remaining 23 detected coordinates C2, C3 . . . C24, subtract 1 if the corresponding pattern bit is a zero or add 1 if it is a one. 
     Step 9: If FLIP=0, ignore this step. Otherwise the detected coordinate identified by FLIP must be increased by 4 if its integer part is 0 or 1 mod 4, or decreased by 4 if it is 2 or 3 mod 4. 
     Step 10: Round each coordinate to the nearest 0 mod 4 value. A coordinate which is exactly 2 mod 4 should be rounded up. The 24 coordinates are now those of the Z lattice point. 
     
                       TABLE 1                                                     
______________________________________                                    
DETECTION ALGORITHM                                                       
In   k      M(n,k)     M(n,k)  P(n,k)   B(n,k)                            
______________________________________                                    
0    0      X(n)**2    1       16-8X(n) 0                                 
1    0      X(n)**2    2+2F(n) 16-8X(n) 0                                 
2    0      (X(n)-4)**2                                                   
                       5-4F(n) 8X(n)-16 1                                 
3    0      (X(n)-4)**2                                                   
                       2-2F(n) 8X(n)-16 1                                 
0    1      (X(n)-1)**2                                                   
                       2-2F(n) 8+8X(n)  0                                 
1    1      (X(n)-1)**2                                                   
                       1       24-8X(n) 0                                 
2    1      (X(n)-1)**2                                                   
                       2+2F(n) 24-8X(n) 0                                 
3    1      (X(n)-5)**2                                                   
                       5-4F(n) 8X(n)-24 1                                 
0    2      (X(n)-2)**2                                                   
                       5-4F(n) 8X(n)    0                                 
1    2      (X(n)-2)**2                                                   
                       2-2F(n) 8X(n)    0                                 
2    2      (X(n)-2)**2                                                   
                       1       32-8X(n) 0                                 
3    2      (X(n)-2)**2                                                   
                       2+2F(n) 32-8X(n) 0                                 
0    3      (X(n)+1)**2                                                   
                       2+2F(n) 8-8X(n)  1                                 
1    3      (X(n)-3)**2                                                   
                       5-4F(n) 8X(n)-8  0                                 
2    3      (X(n)-3)**2                                                   
                       2-2F(n) 8X(n)-8  0                                 
3    3      (X(n)-3)**2                                                   
                       1       40-8X(n) 0                                 
______________________________________                                    
 
    
     
                                           TABLE 2                                 
__________________________________________________________________________
DETECTION ALGORITHM                                                       
J  k1   k2   J  k1   k2   J  k1   k2   J  k1   k2                         
__________________________________________________________________________
000                                                                       
   00000000                                                               
        22222222                                                          
             040                                                          
                02220200                                                  
                     20002022                                             
                          100                                             
                             11111111                                     
                                  33333333                                
                                       140                                
                                          13331311                        
                                               31113133                   
001                                                                       
   00002222                                                               
        22220000                                                          
             041                                                          
                02222022                                                  
                     20000200                                             
                          101                                             
                             11113333                                     
                                  33331111                                
                                       141                                
                                          13333133                        
                                               31111311                   
002                                                                       
   00220022                                                               
        22002200                                                          
             042                                                          
                02000222                                                  
                     20222000                                             
                          102                                             
                             11331133                                     
                                  33113311                                
                                       142                                
                                          13111333                        
                                               31333111                   
003                                                                       
   00222200                                                               
        22000022                                                          
             043                                                          
                02002000                                                  
                     20220222                                             
                          103                                             
                             11333311                                     
                                  33111133                                
                                       143                                
                                          13113111                        
                                               31331333                   
004                                                                       
   02020202                                                               
        20202020                                                          
             044                                                          
                00200002                                                  
                     22022220                                             
                          104                                             
                             13131313                                     
                                  31313131                                
                                       144                                
                                          11311113                        
                                               33133331                   
005                                                                       
   02022020                                                               
        20200202                                                          
             045                                                          
                00202220                                                  
                     22020002                                             
                          105                                             
                             13133131                                     
                                  31311313                                
                                       145                                
                                          11313331                        
                                               33131113                   
006                                                                       
   02200220                                                               
        20022002                                                          
             046                                                          
                00020020                                                  
                     22202202                                             
                          106                                             
                             13311331                                     
                                  31133113                                
                                       146                                
                                          11131131                        
                                               33313313                   
007                                                                       
   02202002                                                               
        20020220                                                          
             047                                                          
                00022202                                                  
                     22200020                                             
                          107                                             
                             13313113                                     
                                  31131331                                
                                       147                                
                                          11133313                        
                                               33311131                   
010                                                                       
   02002202                                                               
        20220020                                                          
             050                                                          
                00222002                                                  
                     22000220                                             
                          110                                             
                             13113313                                     
                                  31331131                                
                                       150                                
                                          11333113                        
                                               33111331                   
011                                                                       
   02000020                                                               
        20222202                                                          
             051                                                          
                00220220                                                  
                     22002002                                             
                          111                                             
                             13111131                                     
                                  31333313                                
                                       151                                
                                          11331331                        
                                               33113113                   
012                                                                       
   02222220                                                               
        20000002                                                          
             052                                                          
                00002020                                                  
                     22220202                                             
                          112                                             
                             13333331                                     
                                  31111113                                
                                       152                                
                                          11113131                        
                                               33331313                   
013                                                                       
   02220002                                                               
        20002220                                                          
             053                                                          
                00000202                                                  
                     22222020                                             
                          113                                             
                             13331113                                     
                                  31113331                                
                                       153                                
                                          11111313                        
                                               33333131                   
014                                                                       
   00022000                                                               
        22200222                                                          
             054                                                          
                02202200                                                  
                     20020022                                             
                          114                                             
                             11133111                                     
                                  33311333                                
                                       154                                
                                          13313311                        
                                               31131133                   
015                                                                       
   00020222                                                               
        22202000                                                          
             055                                                          
                02200022                                                  
                     20022200                                             
                          115                                             
                             11131333                                     
                                  33313111                                
                                       155                                
                                          13311133                        
                                               31133311                   
016                                                                       
   00202022                                                               
        22020200                                                          
             056                                                          
                02022222                                                  
                     20200000                                             
                          116                                             
                             11313133                                     
                                  33131311                                
                                       156                                
                                          13133333                        
                                               31311111                   
017                                                                       
   00200200                                                               
        22022022                                                          
             057                                                          
                02020000                                                  
                     20202222                                             
                          117                                             
                             11311311                                     
                                  33133133                                
                                       157                                
                                          13131111                        
                                               31313333                   
020                                                                       
   02020022                                                               
        20202200                                                          
             060                                                          
                00200222                                                  
                     22022000                                             
                          120                                             
                             13131133                                     
                                  31313311                                
                                       160                                
                                          11311333                        
                                               33133111                   
021                                                                       
   02022200                                                               
        20200022                                                          
             061                                                          
                00202000                                                  
                     22020222                                             
                          121                                             
                             13133311                                     
                                  31311133                                
                                       161                                
                                          11313111                        
                                               33131333                   
022                                                                       
   02200000                                                               
        20022222                                                          
             062                                                          
                00020200                                                  
                     22202022                                             
                          122                                             
                             13311111                                     
                                  31133333                                
                                       162                                
                                          11131311                        
                                               33313133                   
023                                                                       
   02202222                                                               
        20020000                                                          
             063                                                          
                00022022                                                  
                     22200200                                             
                          123                                             
                             13313333                                     
                                  31131111                                
                                       163                                
                                          11133133                        
                                               33311311                   
024                                                                       
   00000220                                                               
        22222002                                                          
             064                                                          
                02220020                                                  
                     20002202                                             
                          124                                             
                             11111331                                     
                                  33333113                                
                                       164                                
                                          13331131                        
                                               31113313                   
025                                                                       
   00002002                                                               
        22220220                                                          
             065                                                          
                02222202                                                  
                     20000020                                             
                          125                                             
                             11113113                                     
                                  33331331                                
                                       165                                
                                          13333313                        
                                               31111131                   
026                                                                       
   00220202                                                               
        22002020                                                          
             066                                                          
                02000002                                                  
                     20222220                                             
                          126                                             
                             11331313                                     
                                  33113131                                
                                       166                                
                                          13111113                        
                                               31333331                   
027                                                                       
   00222020                                                               
        22000202                                                          
             067                                                          
                02002220                                                  
                     20220002                                             
                          127                                             
                             11333131                                     
                                  33111313                                
                                       167                                
                                          13113331                        
                                               31331113                   
030                                                                       
   00022220                                                               
        22200002                                                          
             070                                                          
                02202020                                                  
                     20020202                                             
                          130                                             
                             11133331                                     
                                  33311113                                
                                       170                                
                                          13313131                        
                                               31131313                   
031                                                                       
   00020002                                                               
        22202220                                                          
             071                                                          
                02200202                                                  
                     20022020                                             
                          131                                             
                             11131113                                     
                                  33313331                                
                                       171                                
                                          13311313                        
                                               31133131                   
032                                                                       
   00202202                                                               
        22020020                                                          
             072                                                          
                02022002                                                  
                     20200220                                             
                          132                                             
                             11313313                                     
                                  33131131                                
                                       172                                
                                          13133113                        
                                               31311331                   
033                                                                       
   00200020                                                               
        22022202                                                          
             073                                                          
                02020220                                                  
                     20202002                                             
                          133                                             
                             11311131                                     
                                  33133313                                
                                       173                                
                                          13131331                        
                                               31313113                   
034                                                                       
   02002022                                                               
        20220200                                                          
             074                                                          
                00222222                                                  
                     22000000                                             
                          134                                             
                             13113133                                     
                                  31331311                                
                                       174                                
                                          11333333                        
                                               33111111                   
035                                                                       
   02000200                                                               
        20222022                                                          
             075                                                          
                00220000                                                  
                     22002222                                             
                          135                                             
                             13111311                                     
                                  31333133                                
                                       175                                
                                          11331111                        
                                               33113333                   
036                                                                       
   02222000                                                               
        20000222                                                          
             076                                                          
                00002200                                                  
                     22220022                                             
                          136                                             
                             13333111                                     
                                  31111333                                
                                       176                                
                                          11113311                        
                                               33331133                   
037                                                                       
   02220222                                                               
        20002000                                                          
             077                                                          
                00000022                                                  
                     22222200                                             
                          137                                             
                             13331333                                     
                                  31113111                                
                                       177                                
                                          11111133                        
                                               33333311                   
__________________________________________________________________________
 
    
     
                                           TABLE 3                                 
__________________________________________________________________________
DETECTION ALGORITHM                                                       
j3                                                                        
  j1                                                                      
    j2                                                                    
      bits                                                                
         j1                                                               
           j2                                                             
             bits                                                         
                j1                                                        
                  j2                                                      
                    bits                                                  
                       j1                                                 
                         j2                                               
                           bits                                           
                              j1                                          
                                j2                                        
                                  bits                                    
                                     j1                                   
                                       j2                                 
                                         bits                             
                                            j1                            
                                              j2                          
                                                bits                      
                                                    j1                    
                                                     j2                   
                                                       bits               
__________________________________________________________________________
00                                                                        
  00                                                                      
    00                                                                    
      000                                                                 
         01                                                               
           01                                                             
             100                                                          
                02                                                        
                  02                                                      
                    200                                                   
                       03                                                 
                         03                                               
                           300                                            
                              04                                          
                                04                                        
                                  400                                     
                                     05                                   
                                       05                                 
                                         500                              
                                            06                            
                                              06                          
                                                600                       
                                                   07                     
                                                     07                   
                                                       700                
01                                                                        
  00                                                                      
    01                                                                    
      001                                                                 
         01                                                               
           00                                                             
             101                                                          
                02                                                        
                  03                                                      
                    201                                                   
                       03                                                 
                         02                                               
                           301                                            
                              04                                          
                                05                                        
                                  401                                     
                                     05                                   
                                       04                                 
                                         501       06                     
                                                   07                     
                                                   601                    
                                                   07                     
                                                     06                   
                                                       701                
02                                                                        
  00                                                                      
    02                                                                    
      002                                                                 
         01                                                               
           03                                                             
             102                                                          
                02                                                        
                  00                                                      
                    202                                                   
                       03                                                 
                         01                                               
                           302                                            
                              04                                          
                                06                                        
                                  402                                     
                                     05                                   
                                       07                                 
                                         502       06                     
                                                   04                     
                                                   602                    
                                                   07                     
                                                     05                   
                                                       702                
03                                                                        
  00                                                                      
    03                                                                    
      003                                                                 
         01                                                               
           02                                                             
             103                                                          
                02                                                        
                  01                                                      
                    203                                                   
                       03                                                 
                         00                                               
                           303                                            
                              04                                          
                                07                                        
                                  403                                     
                                     05                                   
                                       06                                 
                                         503       06                     
                                                   05                     
                                                   603                    
                                                   07                     
                                                     04                   
                                                       703                
04                                                                        
  00                                                                      
    04                                                                    
      004                                                                 
         01                                                               
           05                                                             
             104                                                          
                02                                                        
                  06                                                      
                    204                                                   
                       03                                                 
                         07                                               
                           304                                            
                              04                                          
                                00                                        
                                  404                                     
                                     05                                   
                                       01                                 
                                         504       06                     
                                                   02                     
                                                   604                    
                                                   07                     
                                                     03                   
                                                       704                
05                                                                        
  00                                                                      
    05                                                                    
      005                                                                 
         01                                                               
           04                                                             
             105                                                          
                02                                                        
                  07                                                      
                    205                                                   
                       03                                                 
                         06                                               
                           305                                            
                              04                                          
                                01                                        
                                  405                                     
                                     05                                   
                                       00                                 
                                         505       06                     
                                                   03                     
                                                   605                    
                                                   07                     
                                                     02                   
                                                       705                
06                                                                        
  00                                                                      
    06                                                                    
      006                                                                 
         01                                                               
           07                                                             
             106                                                          
                02                                                        
                  04                                                      
                    206                                                   
                       03                                                 
                         05                                               
                           306                                            
                              04                                          
                                02                                        
                                  406                                     
                                     05                                   
                                       03                                 
                                         506       06                     
                                                   00                     
                                                   606                    
                                                   07                     
                                                     01                   
                                                       706                
07                                                                        
  00                                                                      
    07                                                                    
      007                                                                 
         01                                                               
           06                                                             
             107                                                          
                02                                                        
                  05                                                      
                    207                                                   
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  10                                                                      
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34                                                                        
  30                                                                      
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      034                                                                 
         31                                                               
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                         37                                               
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  30                                                                      
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57                                                                        
  50                                                                      
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                         67                                               
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65                                                                        
  60                                                                      
    65                                                                    
      065                                                                 
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                         65                                               
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67                                                                        
  60                                                                      
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      067                                                                 
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                    270                                                   
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                         73                                               
                           370                                            
                              74                                          
                                74                                        
                                  470                                     
                                     75                                   
                                       75                                 
                                         570       76                     
                                                   76                     
                                                   670                    
                                                   77                     
                                                     77                   
                                                       770                
71                                                                        
  70                                                                      
    71                                                                    
      071                                                                 
         71                                                               
           70                                                             
             171                                                          
                72                                                        
                  73                                                      
                    271                                                   
                       73                                                 
                         72                                               
                           371                                            
                              74                                          
                                75                                        
                                  471                                     
                                     75                                   
                                       74                                 
                                         571       76                     
                                                   77                     
                                                   671                    
                                                   77                     
                                                     76                   
                                                       771                
72                                                                        
  70                                                                      
    72                                                                    
      072                                                                 
         71                                                               
           73                                                             
             172                                                          
                72                                                        
                  70                                                      
                    272                                                   
                       73                                                 
                         71                                               
                           372                                            
                              74                                          
                                76                                        
                                  472                                     
                                     75                                   
                                       77                                 
                                         572       76                     
                                                   74                     
                                                   672                    
                                                   77                     
                                                     75                   
                                                       772                
73                                                                        
  70                                                                      
    73                                                                    
      073                                                                 
         71                                                               
           72                                                             
             173                                                          
                72                                                        
                  71                                                      
                    273                                                   
                       73                                                 
                         70                                               
                           373                                            
                              74                                          
                                77                                        
                                  473                                     
                                     75                                   
                                       76                                 
                                         573       76                     
                                                   75                     
                                                   673                    
                                                   77                     
                                                     74                   
                                                       773                
74                                                                        
  70                                                                      
    74                                                                    
      074                                                                 
         71                                                               
           75                                                             
             174                                                          
                72                                                        
                  76                                                      
                    274                                                   
                       73                                                 
                         77                                               
                           374                                            
                              74                                          
                                70                                        
                                  474                                     
                                     75                                   
                                       71                                 
                                         574       76                     
                                                   72                     
                                                   674                    
                                                   77                     
                                                     73                   
                                                       774                
75                                                                        
  70                                                                      
    75                                                                    
      075                                                                 
         71                                                               
           74                                                             
             175                                                          
                72                                                        
                  77                                                      
                    275                                                   
                       73                                                 
                         76                                               
                           375                                            
                              74                                          
                                71                                        
                                  475                                     
                                     75                                   
                                       70                                 
                                         575       76                     
                                                   73                     
                                                   675                    
                                                   77                     
                                                     72                   
                                                       775                
76                                                                        
  70                                                                      
    76                                                                    
      076                                                                 
         71                                                               
           77                                                             
             176                                                          
                72                                                        
                  74                                                      
                    276                                                   
                       73                                                 
                         75                                               
                           376                                            
                              74                                          
                                72                                        
                                  476                                     
                                     75                                   
                                       73                                 
                                         576       76                     
                                                   70                     
                                                   676                    
                                                   77                     
                                                     71                   
                                                       776                
77                                                                        
  70                                                                      
    77                                                                    
      077                                                                 
         71                                                               
           76                                                             
             177                                                          
                72                                                        
                  75                                                      
                    277                                                   
                       73                                                 
                         74                                               
                           377                                            
                              74                                          
                                73                                        
                                  477                                     
                                     75                                   
                                       72                                 
                                         577       76                     
                                                   71                     
                                                   677                    
                                                   77                     
                                                     70                   
                                                       777                
__________________________________________________________________________
 
    
     DECODING ALGORITHM 
     Step 1: Generate S1(1) to S1(24) and R1(1) to R(1)24 from the 24 coordinates of the Z lattice point. Divide each coordinate by 4 and square the result to produce the corresponding S value. Set each R=0 if the corresponding coordinate is positive or zero and set R=1 if it is negative. 
     Step 2: Set I=S1(1), J=S1(2), R=R1(1) and Q=R(1)2. Then use the combining algorithm (described later) with Table A=table Z1 and Table B=table Z1. Let R2A=X and S2A=J 
     Step 3: Set I=S1(3), J=S1(4), R=R(1)3 and Q=R(1)4. Then use the combining algorithm with Table A=table Z1 and Table B=table Z1. Let R2B=X and S2B=J 
     Step 4: Set I=S1(5), J=S1(6), R=R(1)5 and Q=R1(6). Then use the combining algorithm with Table A=table Z1 and Table B=table Z1. Let R2C=X and S2C=J 
     Step 5: Set I=S1(7), J=S1(8), R=R1(7) and Q=R1(8). Then use the combining algorithm with Table A=table Z1 and Table B=table Z1. Let R2D=X and S2D=J 
     Step 6: Set I=S1(9), J=S1(10), R=R1(9) and Q=R1(10). Then use the combining algorithm with Table A=table Z1 and Table B=table Z1. Let R2E=X and S2E=J 
     Step 7: Set I=S1(11), J=S1(12), R=R1(11) and Q=R1(12). Then use the combining algorithm with Table A=table Z1 and Table B=table Z1. Let R2F=X and S2F=J 
     Step 8: Set I=S1(13), J=S1(14), R=R1(13) and Q=R1(14). Then use the combining algorithm with Table A=table Z1 and Table B=table Z1. Let R2G=X and S2G=J 
     Step 9: Set I=S1(15), J=S1(16), R=R1(15) and Q=R1(16). Then use the combining algorithm with Table A=table Z1 and Table B=table Z1. Let R2H=X and S2H=J 
     Step 10: Set I=S1(17), J=S1(18), R=R1(17) and Q=R1(18). Then use the combining algorithm with Table A=table Z1 and Table B=table Z1. Let R2J=X and S2J=J 
     Step 11: Set I=S1(19), J=S1(20), R=R1(19) and Q=R1(20). Then use the combining algorithm with Table A=table Z1 and Table B=table Z1. Let R2K=X and S2K=J 
     Step 12: Set I=S1(21), J=S1(22), R=R1(21) and Q=R1(22). Then use the combining algorithm with Table A=table Z1 and Table B=table Z1. Let R2L=X and S2L=J 
     Step 13: Set I=S1(23), J=S1(24), R=R1(23) and Q=R1(24). Then use the combining algorithm with Table A=table Z1 and Table B=table Z1. Let R2M=X and S2M=J 
     Step 14: Set I=S2B, J=S2C, R=R2B and Q=R2C. Then use the combining algorithm with Table A=table Z2 and Table B=table Z2. Let R4B=X and S4B=J. 
     Step 15: Set I=S2A, J=S4B, R=R2A and Q=R4B. Then use the combining algorithm with Table A=table Z2 and Table B=table Z4. Let R6A=X and S6A=J. 
     Step 16: Set I=S2E, J=S2F, R=R2E and Q=R2F. Then use the combining algorithm with Table A=table Z2 and Table B=table Z2. Let R4E=X and S4E=J. 
     Step 17: Set I=S2D, J=S4E, R=R2D and Q=R4E. Then use the combining algorithm with Table A=table Z2 and Table B=table Z4. Let R6D=X and S6D=J. 
     Step 18: Set I=S2H, J=S2J, R=R2H and Q=R2J. Then use the combining algorithm with Table A=table Z2 and Table B=table Z2. Let R4H=X and S4H=J. 
     Step 19: Set I=S2G, J=S4H, R=R2G and Q=R4H. Then use the combining algorithm with Table A=table Z2 and Table B=table Z4. Let R6G=X and S6G=J. 
     Step 20: Set I=S2L, J=S2M, R=R2L and Q=R2M. Then use the combining algorithm with Table A=table Z2 and Table B=table Z2. Let R4L=X and S4L=J. 
     Step 21: Set I=S2K, J=S4L, R=R2K and Q=R4L. Then use the combining algorithm with Table A=table Z2 and Table B=table Z4. Let R6K=X and S6K=J. 
     Step 22: Set I=S6A, J=S6D, R=R6A and Q=R6D. Then use the combining algorithm with Table A=table Z6 and Table B=table Z6. Let R12A=X and S12A=J. 
     Step 23: Set I=S6G, J=S6K, R=R6G and Q=R6K. Then use the combining algorithm with Table A=table Z6 and Table B=table Z6. Let R12G=X and S12G=J. 
     Step 24: Set I=S12A, J=S12G, R=R12A and Q=R12G. Then use the combining algorithm with Table A=table Z12 and Table B=table Z12. Let D=X and S=J. 
     Step 25: Scan the offset table (table 1) to find the entry for shell S. Add the value in the offset column to D giving a sum M. 
     Step 26: Multiply M by 8192 and add C. The result is the value of N. 
     The combining algorithm invoked above is as follows. 
     Step 1: Multiply Q by entry I of table A and add R to produce X. 
     Step 2: If I=0, the algorithm is complete. Otherwise subtract 1 from I, add 1 to J and then multiply entry I of table A by entry J of table B and add the product to X and repeat this step. 
     
         ______________________________________                                    
ENCODE, DECODE ALGORITHMS                                                 
Offset Table (table 1)                                                    
Shell     Offset                                                          
______________________________________                                    
2         0                                                               
4         1,104                                                           
6         171,168                                                         
8         8,833,888                                                       
10        203,916,208                                                     
12        2,523,373,840                                                   
14        19,754,483,664                                                  
16        113,458,073,424                                                 
18        520,946,091,936                                                 
20        2,008,233,058,864                                               
22        6,753,012,488,080                                               
24        20,276,772,491,728                                              
26        55,527,493,578,896                                              
28        140,471,654,812,016                                             
30        332,599,107,074,288                                             
32        742,583,618,782,064                                             
34        1,577,241,826,163,648                                           
36        3,201,652,977,773,408                                           
38        6,250,829,856,860,656                                           
40        11,772,243,378,517,936                                          
42        21,488,816,280,661,264                                          
44        38,091,349,398,269,712                                          
46        65,813,944,996,460,880                                          
48        110,975,216,780,648,400                                         
50        183,170,699,359,583,888                                         
52        296,176,250,747,821,760                                         
54        470,313,032,057,139,104                                         
56        733,802,649,961,852,704                                         
58        1,127,282,425,012,264,608                                       
60        1,705,562,004,748,326,528                                       
62        2,546,027,906,352,742,656                                       
64        3,750,338,176,583,741,184                                       
66        5,459,698,938,503,358,288                                       
68        7,855,306,052,010,638,928                                       
70        11,185,364,161,430,427,120                                      
72        15,761,598,618,153,912,048                                      
74        22,006,333,253,178,287,488                                      
76        30,439,267,863,141,761,248                                      
78        41,758,145,964,051,168,928                                      
80        56,805,942,222,418,254,624                                      
82        76,705,521,974,670,315,648                                      
84        102,790,003,409,884,499,616                                     
86        136,825,213,243,612,237,984                                     
88        180,871,853,681,611,950,304                                     
90        237,647,674,612,416,058,144                                     
92        310,274,972,509,704,813,568                                     
94        402,855,562,706,550,178,432                                     
96        520,030,903,624,780,216,192                                     
98        667,887,282,878,315,058,752                                     
100       853,204,877,392,029,630,224                                     
102       1,084,866,366,316,724,513,600                                   
104       1,372,628,843,781,043,137,728                                   
106       1,729,261,230,348,138,347,936                                   
108       2,168,594,752,191,117,510,656                                   
110       2,708,748,750,998,791,808,896                                   
112       3,369,061,029,531,845,159,680                                   
114       4,174,907,475,749,930,441,344                                   
116       5,153,012,845,572,620,444,544                                   
118       6,338,485,997,759,954,228,064                                   
120       7,767,814,007,243,823,839,904                                   
122       9,489,088,807,416,724,301,088                                   
124       11,551,560,348,210,870,663,936                                  
126       14,020,397,101,121,777,789,184                                  
128       16,961,486,827,188,962,520,704                                  
130       20,462,257,878,140,766,824,144                                  
132       24,609,951,889,446,752,483,600                                  
134       29,520,947,360,446,140,239,888                                  
136       35,309,668,202,124,931,709,648                                  
138       42,129,631,517,897,299,694,192                                  
140       50,129,857,007,333,339,232,368                                  
142       59,511,140,048,476,876,253,552                                  
144       70,465,868,440,255,241,753,840                                  
146       83,255,084,071,227,534,308,224                                  
148       98,125,093,343,012,843,683,744                                  
150       115,412,613,794,490,713,387,776                                 
152       135,431,325,977,840,557,716,032                                 
154       158,612,397,740,580,988,792,832                                 
156       185,352,173,129,019,034,813,952                                 
158       216,200,163,114,557,452,649,088                                 
160       251,653,554,301,571,825,870,208                                 
162       292,407,894,227,049,291,537,120                                 
164       339,084,303,287,704,694,137,264                                 
166       392,557,502,327,740,879,513,552                                 
168       453,598,102,325,630,140,035,472                                 
170       523,302,245,924,753,258,804,112                                 
172       602,619,554,566,350,712,176,720                                 
174       692,915,189,017,751,886,612,624                                 
176       795,356,236,008,523,094,321,424                                 
178       911,633,121,557,068,121,682,768                                 
180       1,043,170,702,590,018,684,540,528                               
182       1,192,056,699,908,246,215,817,040                               
184       1,360,020,268,709,003,330,894,928                               
186       1,549,625,404,920,394,770,580,176                               
188       1,762,966,570,548,122,052,971,216                               
190       2,003,176,071,047,978,658,185,168                               
192       2,272,775,796,193,092,396,592,208                               
194       2,575,585,656,054,138,708,462,352                               
196       2,914,624,028,838,764,473,032,112                               
198       3,294,525,191,675,170,663,348,864                               
200       3,718,899,779,744,493,071,204,800                               
202       4,193,342,739,035,337,171,766,768                               
204       4,722,146,931,451,409,521,239,376                               
206       5,312,060,154,473,602,161,379,024                               
208       5,968,158,963,699,136,006,897,744                               
210       6,698,542,086,960,014,568,788,272                               
212       7,509,212,883,615,628,699,979,056                               
214       8,409,846,821,357,616,140,398,480                               
216       9,407,504,001,506,860,931,212,240                               
218       10,513,739,929,400,091,829,175,120                              
220       11,736,809,764,883,851,223,578,160                              
222       13,090,450,253,047,376,115,538,352                              
224       14,584,327,787,390,434,105,633,840                              
226       16,234,701,307,190,010,119,347,120                              
228       18,052,821,776,055,874,414,204,240                              
______________________________________                                    
 
    
     
                       TABLE Z1                                                    
______________________________________                                    
0     1     46     0   92    0    138  0   184   0                        
1     2     47     0   93    0    139  0   185   0                        
2     0     48     0   94    0    140  0   186   0                        
3     0     49     2   95    0    141  0   187   0                        
4     2     50     0   96    0    142  0   188   0                        
5     0     51     0   97    0    143  0   189   0                        
6     0     52     0   98    0    144  2   190   0                        
7     0     53     0   99    0    145  0   191   0                        
8     0     54     0   100   2    146  0   192   0                        
9     2     55     0   101   0    147  0   193   0                        
10    0     56     0   102   0    148  0   194   0                        
11    0     57     0   103   0    149  0   195   0                        
12    0     58     0   104   0    150  0   196   2                        
13    0     59     0   105   0    151  0   197   0                        
14    0     60     0   106   0    152  0   198   0                        
15    0     61     0   107   0    153  0   199   0                        
16    2     62     0   108   0    154  0   200   0                        
17    0     63     0   109   0    155  0   201   0                        
18    0     64     2   110   0    156  0   202   0                        
19    0     65     0   111   0    157  0   203   0                        
20    0     66     0   112   0    158  0   204   0                        
21    0     67     0   113   0    159  0   205   0                        
22    0     68     0   114   0    160  0   206   0                        
23    0     69     0   115   0    161  0   207   0                        
24    0     70     0   116   0    162  0   208   0                        
25    2     71     0   117   0    163  0   209   0                        
26    0     72     0   118   0    164  0   210   0                        
27    0     73     0   119   0    165  0   211   0                        
28    0     74     0   120   0    166  0   212   0                        
29    0     75     0   121   2    167  0   213   0                        
30    0     76     0   122   0    168  0   214   0                        
31    0     77     0   123   0    169  2   215   0                        
32    0     78     0   124   0    170  0   216   0                        
33    0     79     0   125   0    171  0   217   0                        
34    0     80     0   126   0    172  0   218   0                        
35    0     81     2   127   0    173  0   219   0                        
36    2     82     0   128   0    174  0   220   0                        
37    0     83     0   129   0    175  0   221   0                        
38    0     84     0   130   0    176  0   222   0                        
39    0     85     0   131   0    177  0   223   0                        
40    0     86     0   132   0    178  0   224   0                        
41    0     87     0   133   0    179  0   225   2                        
42    0     88     0   134   0    180  0   226   0                        
43    0     89     0   135   0    181  0   227   0                        
44    0     90     0   136   0    182  0   228   0                        
45    0     91     0   137   0    183  0                                  
______________________________________                                    
 
    
     
                       TABLE Z2                                                    
______________________________________                                    
0     1     46     0   92    0   138   0   184   0                        
1     4     47     0   93    0   139   0   185   16                       
2     4     48     0   94    0   140   0   186   0                        
3     0     49     4   95    0   141   0   187   0                        
4     4     50     12  96    0   142   0   188   0                        
5     8     51     0   97    8   143   0   189   0                        
6     0     52     8   98    4   144   4   190   0                        
7     0     53     8   99    0   145   16  191   0                        
8     4     54     0   100   12  146   8   192   0                        
9     4     55     0   101   8   147   0   193   8                        
10    8     56     0   102   0   148   8   194   8                        
11    0     57     0   103   0   149   8   195   0                        
12    0     58     8   104   8   150   0   196   4                        
13    8     59     0   105   0   151   0   197   8                        
14    0     60     0   106   8   152   0   198   0                        
15    0     61     8   107   0   153   8   199   0                        
16    4     62     0   108   0   154   0   200   12                       
17    8     63     0   109   8   155   0   201   0                        
18    4     64     4   110   0   156   0   202   8                        
19    0     65     16  111   0   157   8   203   0                        
20    8     66     0   112   0   158   0   204   0                        
21    0     67     0   113   8   159   0   205   16                       
22    0     68     8   114   0   160   8   206   0                        
23    0     69     0   115   0   161   0   207   0                        
24    0     70     0   116   8   162   4   208   8                        
25    12    71     0   117   8   163   0   209   0                        
26    8     72     4   118   0   164   8   210   0                        
27    0     73     8   119   0   165   0   211   0                        
28    0     74     8   120   0   166   0   212   8                        
29    8     75     0   121   4   167   0   213   0                        
30    0     76     0   122   8   168   0   214   0                        
31    0     77     0   123   0   169   12  215   0                        
32    4     78     0   124   0   170   16  216   0                        
33    0     79     0   125   16  171   0   217   0                        
34    8     80     8   126   0   172   0   218   8                        
35    0     81     4   127   0   173   8   219   0                        
36    4     82     8   128   4   174   0   220   0                        
37    8     83     0   129   0   175   0   221   16                       
38    0     84     0   130   16  176   0   222   0                        
39    0     85     16  131   0   177   0   223   0                        
40    8     86     0   132   0   178   8   224   0                        
41    8     87     0   133   0   179   0   225   12                       
42    0     88     0   134   0   180   8   226   8                        
43    0     89     8   135   0   181   8   227   0                        
44    0     90     8   136   8   182   0   228   0                        
45    8     91     0   137   8   183   0                                  
______________________________________                                    
 
    
     
                       TABLE Z4                                                    
______________________________________                                    
0     1      58      720  116    720  174   2,880                         
1     8      59      480  117    1,456                                    
                                      175   1,984                         
2     24     60      576  118    1,440                                    
                                      176   288                           
3     32     61      496  119    1,152                                    
                                      177   1,920                         
4     24     62      768  120    576  178   2,160                         
5     48     63      832  121    1,064                                    
                                      179   1,440                         
6     96     64      24   122    1,488                                    
                                      180   1,872                         
7     64     65      672  123    1,344                                    
                                      181   1,456                         
8     24     66      1,152                                                
                          124    768  182   2,688                         
9     104    67      544  125    1,248                                    
                                      183   1,984                         
10    144    68      432  126    2,496                                    
                                      184   576                           
11    96     69      768  127    1,024                                    
                                      185   1,824                         
12    96     70      1,152                                                
                          128    24   186   3,072                         
13    112    71      576  129    1,408                                    
                                      187   1,728                         
14    192    72      312  130    2,016                                    
                                      188   1,152                         
15    192    73      592  131    1,056                                    
                                      189   2,560                         
16    24     74      912  132    1,152                                    
                                      190   2,880                         
17    144    75      992  133    1,280                                    
                                      191   1,536                         
18    312    76      480  134    1,632                                    
                                      192   96                            
19    160    77      768  135    1,920                                    
                                      193   1,552                         
20    144    78      1,344                                                
                          136    432  194   2,352                         
21    256    79      640  137    1,104                                    
                                      195   2,688                         
22    288    80      144  138    2,304                                    
                                      196   1,368                         
23    192    81      968  139    1,120                                    
                                      197   1,584                         
24    96     82      1,008                                                
                          140    1,152                                    
                                      198   3,744                         
25    248    83      672  141    1,536                                    
                                      199   1,600                         
26    336    84      768  142    1,728                                    
                                      200   744                           
27    320    85      864  143    1,344                                    
                                      201   2,176                         
28    192    86      1,056                                                
                          144    312  202   2,448                         
29    240    87      960  145    1,440                                    
                                      203   1,920                         
30    576    88      288  146    1,776                                    
                                      204   1,728                         
31    256    89      720  147    1,824                                    
                                      205   2,016                         
32    24     90      1,872                                                
                          148    912  206   2,496                         
33    384    91      896  149    1,200                                    
                                      207   2,496                         
34    432    92      576  150    2,976                                    
                                      208   336                           
35    384    93      1,024                                                
                          151    1,216                                    
                                      209   1,920                         
36    312    94      1,152                                                
                          152    480  210   4,608                         
37    304    95      960  153    1,872                                    
                                      211   1,696                         
38    480    96      96   154    2,304                                    
                                      212   1,296                         
39    448    97      784  155    1,536                                    
                                      213   2,304                         
40    144    98      1,368                                                
                          156    1,344                                    
                                      214   2,592                         
41    336    99      1,248                                                
                          157    1,264                                    
                                      215   2,112                         
42    768    100     744  158    1,920                                    
                                      216   960                           
43    352    101     816  159    1,728                                    
                                      217   2,048                         
44    288    102     1,728                                                
                          160    144  218   2,640                         
45    624    103     832  161    1,536                                    
                                      219   2,368                         
46    576    104     336  162    2,904                                    
                                      220   1,728                         
47    384    105     1,536                                                
                          163    1,312                                    
                                      221   2,016                         
48    96     106     1,296                                                
                          164    1,008                                    
                                      222   3,648                         
49    456    107     864  165    2,304                                    
                                      223   1,792                         
50    744    108     960  166    2,016                                    
                                      224   192                           
51    576    109     880  167    1,344                                    
                                      225   3,224                         
52    336    110     1,728                                                
                          168    768  226   2,736                         
53    432    111     1,216                                                
                          169    1,464                                    
                                      227   1,824                         
54    960    112     192  170    2,592                                    
                                      228   1,920                         
55    576    113     912  171    2,080                                    
56    192    114     1,920                                                
                          172    1,056                                    
57    640    115     1,152                                                
                          173    1,392                                    
______________________________________                                    
 
    
     
                       TABLE Z6                                                    
______________________________________                                    
0      1       77       69,120                                            
                              154    345,600                              
1      12      78       92,480                                            
                              155    499,200                              
2      60      79       124,800                                           
                              156    353,600                              
3      160     80       106,392                                           
                              157    295,800                              
4      252     81       70,860                                            
                              158    424,320                              
5      312     82       100,920                                           
                              159    449,600                              
6      544     83       137,760                                           
                              160    425,880                              
7      960     84       96,768                                            
                              161    304,128                              
8      1,020   85       90,480                                            
                              162    354,300                              
9      876     86       125,664                                           
                              163    531,360                              
10     1,560   87       134,720                                           
                              164    423,864                              
11     2,400   88       123,360                                           
                              165    299,520                              
12     2,080   89       95,064                                            
                              166    468,384                              
13     2,040   90       113,880                                           
                              167    557,760                              
14     3,264   91       163,200                                           
                              168    391,680                              
15     4,160   92       137,280                                           
                              169    344,772                              
16     4,092   93       92,160                                            
                              170    452,400                              
17     3,480   94       150,144                                           
                              171    525,600                              
18     4,380   95       187,200                                           
                              172    480,480                              
19     7,200   96       131,104                                           
                              173    359,160                              
20     6,552   97       112,920                                           
                              174    458,048                              
21     4,608   98       141,180                                           
                              175    624,960                              
22     8,160   99       175,200                                           
                              176    492,000                              
23     10,560  100      164,052                                           
                              177    334,080                              
24     8,224   101      122,424                                           
                              178    475,320                              
25     7,812   102      157,760                                           
                              179    640,800                              
26     10,200  103      212,160                                           
                              180    478,296                              
27     13,120  104      173,400                                           
                              181    393,144                              
28     12,480  105      119,808                                           
                              182    554,880                              
29     10,104  106      168,600                                           
                              183    595,520                              
30     14,144  107      228,960                                           
                              184    542,784                              
31     19,200  108      170,560                                           
                              185    427,440                              
32     16,380  109      142,584                                           
                              186    460,800                              
33     11,520  110      212,160                                           
                              187    696,000                              
34     17,400  111      219,200                                           
                              188    574,080                              
35     24,960  112      196,800                                           
                              189    377,856                              
36     18,396  113      153,240                                           
                              190    636,480                              
37     16,440  114      172,800                                           
                              191    729,600                              
38     24,480  115      274,560                                           
                              192    524,320                              
39     27,200  116      212,184                                           
                              193    447,000                              
40     26,520  117      148,920                                           
                              194    564,600                              
41     20,184  118      236,640                                           
                              195    707,200                              
42     23,040  119      278,400                                           
                              196    592,956                              
43     36,960  120      213,824                                           
                              197    465,720                              
44     31,200  121      174,252                                           
                              198    595,680                              
45     22,776  122      223,320                                           
                              199    792,000                              
46     35,904  123      269,120                                           
                              200    664,020                              
47     44,160  124      249,600                                           
                              201    430,848                              
48     32,800  125      195,312                                           
                              202    612,120                              
49     28,236  126      238,272                                           
                              203    808,320                              
50     39,060  127      322,560                                           
                              204    603,200                              
51     46,400  128      262,140                                           
                              205    524,784                              
52     42,840  129      177,408                                           
                              206    721,344                              
53     33,720  130      265,200                                           
                              207    770,880                              
54     44,608  131      343,200                                           
                              208    695,640                              
55     62,400  132      241,920                                           
                              209    518,400                              
56     49,344  133      207,360                                           
                              210    599,040                              
57     34,560  134      305,184                                           
                              211    890,400                              
58     50,520  135      341,120                                           
                              212    708,120                              
59     69,600  136      295,800                                           
                              213    483,840                              
60     54,080  137      225,240                                           
                              214    778,464                              
61     44,664  138      253,440                                           
                              215    960,960                              
62     65,280  139      386,400                                           
                              216    674,368                              
63     70,080  140      324,480                                           
                              217    552,960                              
64     65,532  141      211,968                                           
                              218    712,920                              
65     53,040  142      342,720                                           
                              219    852,800                              
66     57,600  143      408,000                                           
                              220    811,200                              
67     89,760  144      298,716                                           
                              221    591,600                              
68     73,080  145      262,704                                           
                              222    745,280                              
69     50,688  146      319,800                                           
                              223    994,560                              
70     84,864  147      376,480                                           
                              224    786,624                              
71     100,800 148      345,240                                           
                              225    570,276                              
72     74,460  149      266,424                                           
                              226    766,200                              
73     63,960  150      354,144                                           
                              227    1,030,560                            
74     82,200  151      456,000                                           
                              228    725,760                              
75     104,160 152      370,080                                           
76     93,600  153      254,040                                           
______________________________________                                    
 
    
     
                       TABLE Z12                                                   
______________________________________                                    
0   1          77     21,654,935,808                                      
                                154 714,637,972,224                       
1   24         78     23,917,274,304                                      
                                155 715,961,510,400                       
2   264        79     24,615,220,608                                      
                                156 719,692,526,784                       
3   1,760      80     25,376,943,024                                      
                                157 763,113,874,512                       
4   7,944      81     28,009,137,752                                      
                                158 812,342,889,600                       
5   25,872     82     30,586,037,328                                      
                                159 816,313,861,440                       
6   64,416     83     31,513,408,608                                      
                                160 812,060,601,264                       
7   133,056    84     32,579,551,488                                      
                                161 865,462,338,048                       
8   253,704    85     35,508,322,080                                      
                                162 924,314,846,664                       
9   472,760    86     38,810,229,216                                      
                                163 920,507,154,912                       
10  825,264    87     40,037,878,848                                      
                                164 920,361,668,688                       
11  1,297,056  88     40,859,536,608                                      
                                165 982,725,974,784                       
12  1,938,336  89     44,672,951,664                                      
                                166 1,039,906,730,016                     
13  2,963,664  90     48,932,378,352                                      
                                167 1,039,128,015,552                     
14  4,437,312  91     49,926,264,960                                      
                                168 1,040,478,667,008                     
15  6,091,584  92     51,130,316,736                                      
                                169 1,102,867,760,040                     
16  8,118,024  93     55,883,287,552                                      
                                170 1,171,757,692,512                     
17  11,368,368 94     60,547,082,112                                      
                                171 1,174,521,854,176                     
18  15,653,352 95     61,923,031,104                                      
                                172 1,167,835,079,136                     
19  19,822,176 96     63,385,408,416                                      
                                173 1,239,722,456,016                     
20  24,832,944 97     68,696,763,696                                      
                                174 1,321,246,238,400                     
21  32,826,112 98     74,577,903,048                                      
                                175 1,313,545,628,736                     
22  42,517,728 99     76,393,194,528                                      
                                176 1,307,424,001,248                     
23  51,425,088 100    77,602,957,944                                      
                                177 1,395,533,705,856                     
24  61,903,776 101    84,080,984,592                                      
                                178 1,474,191,694,800                     
25  78,146,664 102    91,461,572,928                                      
                                179 1,470,135,873,888                     
26  98,021,616 103    92,741,489,856                                      
                                180 1,472,419,748,592                     
27  115,331,264                                                           
               104    94,198,772,976                                      
                                181 1,554,105,008,016                     
28  133,522,752                                                           
               105    102,562,521,600                                     
                                182 1,647,547,321,728                     
29  164,079,696                                                           
               106    110,403,610,416                                     
                                183 1,648,658,651,200                     
30  201,364,416                                                           
               107    112,206,096,288                                     
                                184 1,632,926,218,176                     
31  229,101,312                                                           
               108    114,458,740,800                                     
                                185 1,734,153,444,192                     
32  259,776,264                                                           
               109    123,091,514,832                                     
                                186 1,844,175,455,232                     
33  314,269,824                                                           
               110    132,910,417,728                                     
                                187 1,829,372,897,088                     
34  374,842,512                                                           
               111    135,359,463,232                                     
                                188 1,821,916,743,552                     
35  420,258,432                                                           
               112    136,447,747,392                                     
                                189 1,937,378,426,368                     
36  471,023,592                                                           
               113    147,394,340,016                                     
                                190 2,043,436,190,400                     
37  554,746,896                                                           
               114    159,500,457,600                                     
                                191 2,033,547,631,104                     
38  653,690,400                                                           
               115    160,956,544,896                                     
                                192 2,028,332,946,336                     
39  724,846,144                                                           
               116    162,940,575,600                                     
                                193 2,142,281,587,248                     
40  793,078,704                                                           
               117    176,121,743,248                                     
                                194 2,267,057,828,112                     
41  927,125,232                                                           
               118    188,740,015,200                                     
                                195 2,265,622,499,712                     
42  1,082,704,128                                                         
               119    190,918,949,760                                     
                                196 2,244,116,900,808                     
43  1,175,873,952                                                         
               120    193,511,203,776                                     
                                197 2,373,665,758,992                     
44  1,279,397,088                                                         
               121    207,502,774,008                                     
                                198 2,521,003,646,304                     
45  1,482,713,808                                                         
               122    222,973,423,728                                     
                                199 2,496,637,706,688                     
46  1,699,194,816                                                         
               123    226,147,998,912                                     
                                200 2,478,371,203,704                     
47  1,834,739,328                                                         
               124    227,429,983,488                                     
                                201 2,635,440,022,912                     
48  1,980,797,856                                                         
               125    244,215,894,432                                     
                                202 2,774,666,532,528                     
49  2,259,791,448                                                         
               126    263,101,540,416                                     
                                203 2,758,012,109,952                     
50  2,578,950,264                                                         
               127    264,312,339,456                                     
                                204 2,752,161,876,288                     
51  2,771,448,768                                                         
               128    266,010,877,704                                     
                                205 2,897,346,782,880                     
52  2,949,559,536                                                         
               129    286,962,805,888                                     
                                206 3,060,483,556,416                     
53  3,345,875,856                                                         
               130    306,415,571,616                                     
                                207 3,053,047,659,072                     
54  3,803,764,800                                                         
               131    308,637,532,896                                     
                                208 3,014,173,603,056                     
55  4,028,054,976                                                         
               132    312,172,889,472                                     
                                209 3,190,254,080,640                     
56  4,264,256,832                                                         
               133    332,945,133,312                                     
                                210 3,384,533,104,128                     
57  4,833,186,688                                                         
               134    356,433,028,512                                     
                                211 3,345,812,189,280                     
58  5,414,943,600                                                         
               135    360,323,811,456                                     
                                212 3,322,145,004,336                     
59  5,719,271,712                                                         
               136    360,223,654,032                                     
                                213 3,521,864,694,528                     
60  6,059,238,336                                                         
               137    386,088,721,392                                     
                                214 3,702,736,569,312                     
61  6,756,214,608                                                         
               138    414,603,535,104                                     
                                215 3,676,376,713,152                     
62  7,558,096,128                                                         
               139    415,108,384,416                                     
                                216 3,655,417,972,800                     
63  7,972,913,344                                                         
               140    417,392,122,752                                     
                                217 3,849,584,302,080                     
64  8,312,839,944                                                         
               141    447,681,704,448                                     
                                218 4,061,967,241,200                     
65  9,284,959,200                                                         
               142    476,316,548,928                                     
                                219 4,046,622,779,584                     
66  10,374,325,632                                                        
               143    478,377,518,784                                     
                                220 3,999,395,297,088                     
67  10,801,349,856                                                        
               144    481,341,997,032                                     
                                221 4,217,474,077,344                     
68  11,279,351,952                                                        
               145    512,942,325,984                                     
                                222 4,466,860,398,528                     
69  12,564,531,456                                                        
               146    547,290,900,816                                     
                                223 4,411,797,088,512                     
70  13,871,037,312                                                        
               147    551,426,235,360                                     
                                224 4,366,319,445,312                     
71  14,433,084,864                                                        
               148    550,868,402,352                                     
                                225 4,633,748,755,400                     
72  15,042,871,272                                                        
               149    587,521,351,824                                     
                                226 4,864,028,873,616                     
73  16,585,653,744                                                        
               150    629,263,864,416                                     
                                227 4,821,900,001,248                     
74  18,306,804,912                                                        
               151    628,022,995,776                                     
                                228 4,799,513,769,600                     
75  19,068,642,080                                                        
               152    628,196,474,400                                     
76  19,670,138,400                                                        
               153    673,500,182,256                                     
______________________________________                                    
 
    
     There has been described a preferred embodiment of an encoding, detecting and decoding method wherein the modified Leech matrix is used, at the encoder to modify the coordinates for transmission which coordinates, so modified, are used in the detection and decoding means. The modified Leech matrix involves both column rearranging and row conversions of the conventional Leech matrix. 
     In an alternate form of the invention, it is possible to use the novel detection methods with received transmissions which have been encoded in accord with other matrices having the same column arrangement as the modified Leech matrix. That is with matrices which have the same column arrangement as the modified matrix but a different row arrangement called here SCDR Matrix for brevity (although any row of a different row arrangement must be derivable from the modified matrix rows by mod 2 sums which include the original modified matrix now). Since all such SCDR matrices will give the same 4096 pattern vectors as the modified matrix all detection may equivalently be used for the encode and decode means with the algorithm as provided herein with one change. The thirteen bits thus derived at the detector identity rows of the modified Leech matrix rather than the matrix actually used at the encoder. Thus these thirteen bits must be converted to thirteen corresponding buts which are the determinant for the SCDR matrix used. Such conversion may be performed by well known methods, such as by the use of a 13×13 binary matrix or by look up table. 
     In a further alternative it is possible to use the novel detection means and method with encoder matrices of the form of the Conventional Leech matrix or derived therefrom (or otherwise using a column arrangement different from that of the modified matrix) column changes used for the new matrix. Such matrices are designated here DC matrices for brevity. Encoding is performed in accord with the DC matrix and this is modulated and transmitted to the receiver. After demodulation the column arrangement is altered to conform to that of a Modified Leech matrix. The newly placed received coordinates are then subjected to the detection algorithms steps 1 to 5 as described herein. With the 13 bits derived it is now necessary to take into account any difference in row conversion. This may be done by mapping the 13 derived bits into a new 13 bits in accord with the use of a 13×13 binary matrix by techniques well known to those skilled in the art. With the new 13 bits rows of the modified matrix may be sumed mod 2 to indicate the pattern vector with the modified matrix place order. With such pattern vector it is possible to do one of two things: (a) to modify the stored altered place coordinate values in accord with steps 7 &amp; 8 and then reverse the alterations in the coordinate places to produce the order used at the transmitter, then continuing with step 9 of the detection algorithm; or (b) reversing the alterations in the places of the pattern vector and using the reversed alteration pattern vector to modify the coordinates as received before alternation then continuing with step 6. 
     It will be noted that the provision in a matrix derived from the conventional Leech matrix by column shifts and by row combining in the ways previously discussed which contain as three of the nine rows: 
     
         ______________________________________                                    
1 1 1 1 1 1 1 1                                                           
              0 0 0 0 0 0 0 0                                             
                         0 0 0 0 0 0 0 0                                  
0 0 0 0 0 0 0 0                                                           
              1 1 1 1 1 1 1 1                                             
                         0 0 0 0 0 0 0 0                                  
0 0 0 0 0 0 0 0                                                           
              0 0 0 0 0 0 0 0                                             
                         1 1 1 1 1 1 1 1                                  
______________________________________                                    
 
    
     where consecutive or not, determines the fact that the octuples produced by pattern vectors derived from the derived matrix may be classified into 64 complementary pairs. 
     ROW ADDITION TABLE 
     This is the extended form of the row transition of the modified Leech matrix. 
     (11,2) means row 11 of the Conventional Leech matrix was added place by place mod 2 to form a new row 2, row 11  remaining as it was. Thus the second step 2, 11 requires new row 2 to be added place by place, mod 2 to form a new row 11. This logic is continued throughout. It is not suggested that this is the shortest or only method of deriving the row arrangement of the modified from the Conventional Leech matrix. Elsewhere discussed, is the columns&#39; rearrangement to get from the conventional to modified Leech matrix. 
     
         ______________________________________                                    
(11,2)   (7,9)     (10,11)   (4,8)   (8,4)                                
(2,11)   (7,10)    (11,10)   (4,12)  (8,7)                                
(3,4)    (7,11)    (10,11)   (6,5)   (8,9)                                
(3,5)    (7,12)    (11,12)   (5,6)   (8,11)                               
(3,6)    (8,7)     (12,11)   (5,4)   (12,10)                              
(3,7)    (8,6)     (11,12)   (5,3)   (10,12)                              
(3,9)    (9,2)     (1,2)     (5,3)   (10,9)                               
(4,5)    (9,3)     (1,4)     (5,8)   (10,7)                               
(4,6)    (10,12)   (1,7)     (5,12)  (10,5)                               
(4,9)    (10,9)    (1,11)    (12,6)  (10,2)                               
(6,5)    (10,8)    (1,12)    (6,5)   (11,10)                              
(5,6)    (10,6)    (2,3)     (6,7)   (11,9)                               
(5,3)    (10,5)    (2,5)     (6,8)   (11,2)                               
(5,9)    (10,2)    (2,6)     (6,11)  (12,11)                              
(5,11)   (7,3)     (2,7)     (6,12)  (12,9)                               
(8,6)    (8,3)     (2,11)    (11,9)  (12,7)                               
(6,8)    (9,3)     (2,12)    (9,2)   (12,6)                               
(6,3)    (11,4)    (4,3)     (9,3)   (12,4)                               
(6,9)    (11,3)    (3,4)     (9,4)   (12,2)                               
(6,10)   (11,2)    (3,2)     (9,7)                                        
(6,11)   (12,11)   (3,6)     (9,10)                                       
(6,12)   (12,10)   (3,7)     (9,11)                                       
(8,7)    (12,9)    (3,9)     (11,7)                                       
(7,8)    (12,8)    (3,11)    (7,11)                                       
(7,2)    (12,7)    (3,12)    (7,9)                                        
(7,3)    (12,6)    (6,4)     (7,3)                                        
         (12,4)    (4,3)     (8,3)                                        
                   (4,6)                                                  
______________________________________                                    
 
    
     As previously stated `mod` is used as an abbreviation for `modulo` herein.