Generalization of Multiplication M-Sequences over Fp and Its Recip- rocal Sequences

Sloane, N.J.A., studied the product or multiplication sequence {zn} on t degrees of the sequence {an} which has the degree of complexity r and gave the answer that the degree of complexity of {zn} can’t be exceeded . Mokayes D. Al Cheikha A. H., studied the construction of the multiplication binary M-Sequences where the multiplication on one M-Sequence or on more than one sequence and gave an illustrated about the question “why the length of the linear equivalent shift register don’t reached the maximum length?” . Al Cheikha A. H., Omar E H.., studied the construction of the multiplication binary M-Sequences with it’s reciprocal sequences, they studied the length of the result multiplication sequence{zn}which is equals [(deg f(x)) deg (f(x))], linear equivalent, properties of the coefficients of the roots of the characteristic polynomial f(x) and it’s reciprocal sequence. Current article and showing the agreement and disagreement between these coefficients and corresponding roots, in the first side, in binary M-Sequences, and in the second side, Mp-Sequences .

is called the initial vector and the characteristic equation of the sequence is; (2) [12][13][14] We are limited in our article to 2 > p .

Definitions and Theorems
Definition 1 The L.F.S.R is a linear feedback shift register if contains only addition circuits and the general term of the sequence {s n } generated through the shift register is the term of the output of the register [3] . Definition 2 The vector ) ,..., , ( , and 1 , ≤ l t R [8][9][10]13] .

Definition 4
The periodic sequence is called an orthogonal set if and only if the set A satisfies the following two conditions; (8) [15,16]

Definition 6
If the function f(x) has the degree n then the reciprocal function of f(x) is the function; and this sequence is called M p -Sequence or M-Sequence over F p [2,[15][16][17] .

Theorem 9
If } { n a is a H.L.R.S in F p and ) (x g is its characteristic prime polynomial of degree k and α is a root of ) (x g in any splitting field of F p then the general term of the sequence } { n a is; And the sequence } { n a has the period 1 − k p [2,18] .
*Our study is limited to the M p -Sequence or M-Sequence over F p and it's period is:

Multiplication Two Reciprocal Mp-Sequences
If f(x) is the characteristic prime polynomial of the M p -Sequence{s n } of degree k in the field F p , and α 1 , α 2 , … α k are it's independent different roots in the F p k , which is a splitting field of F p then the general term of the sequence{s n }is    x x x f which is a prime characteristic polynomial the sequence {s n }, 3 ; F s N n n ∈ ∈ ∀ s n satisfies; We can see the important properties; ( ) ( ) By the same way; We can see the important properties; By the same way; ( ) ( ) The sequence } { n ζ is periodic with the period We can look that one period of the sequence } { n ζ is one period of the sequence {s n }but through reading it by inverse from the right to the left.
The multiplication sequence {z n }, where z n = s n . n ζ is; We can look that one period of the sequence } { n  is one period of the sequence {s n }but through reading it by inverse from the right to the left.
The multiplication sequence {z n }, where z n = s n . n  is; Thus, the sequence {z n } is a linear nonhomogeneous sequence with the length of its linear homogeneous equivalent is equals 2 that is equal , The period of {z n }is 4 and it's the half period of the sequence{s n }and the sequence {z n } is: 0110 0110 0110 0110 …….
We can check that the set of the all periodic permutation of one period is not an orthogonal set for example , for one permutation of the period: 0110 is: 0011 and the sum of the two vectors is 0121 and is not belong to the set of permutations. Figure 3 illustrates the feedback shift registers generating the sequence {z n }; Or; Thus, the sequence {z n } is a linear nonhomogeneous sequence with the length of its linear homogeneous equivalent is equals 2 that is equal to x f x f , The period of {z n }is 4 and it's the half period of the sequence{s n }and the sequence {z n } is: 0110 0110 0110 0110 …….
We can check that the set of the all periodic permutation of one period is not an orthogonal set for example , for one permutation of the period: 0110 is: 0011 and the sum of the two vectors is 0121 and is not belong to the set of permutations. Figure 3 illustrates the feedback shift registers generating the sequence {z n };   Suppose the polynomial which is a prime characteristic polynomial the sequence {a n } as in figure5, 3 ; F a N n n ∈ ∈ ∀ a n satisfies; The general term of the sequence {an}is of the form; 22  3  16  3  3   3  16  3  14  3  2   2  14  3  22  3 And the general solution of the characteristic equation is; And the sequence is periodic with the period: 3 3 -1 = 26 and the sequence is; 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 …. (37) The reciprocal polynomial of the prime f(x) is the prime polynomial The general term of the sequence {a n }is of the form; Solving the following equations ; We can see the properties; And the sequence is periodic with the period: 3 3 -1 = 26 and the sequence is; 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 …. (37) The reciprocal polynomial of the prime f(x) is the prime polynomial Is very easy looking that The general term of the sequence {a n }is of the form; Solving the following equations ; We can see the properties; And the sequence is periodic with the period: 3 3 -1 = 26 and the sequence is; 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 …. (37) The reciprocal polynomial of the prime f(x) is the prime polynomial Is very easy looking that We can see the properties; The general term of the sequence {a n }is of the form; Solving the following equations ; We can see the properties; And the sequence is periodic with the period: 3 3 -1 = 26 and the sequence is; 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 …. (37) The reciprocal polynomial of the prime f(x) is the prime polynomial Is very easy looking that And the general solution of the characteristic equation is; The general term of the sequence {a n }is of the form; Solving the following equations ; We can see the properties; And the sequence is periodic with the period: 3 3 -1 = 26 and the sequence is; 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 …. (37) The reciprocal polynomial of the prime f(x) is the prime polynomial Is very easy looking that The general term of the sequence {a n }is of the form; Solving the following equations ; We can see the properties ;   1  22  3  16  3  3   3  16  3  14  3  2   2  14  3  22  3 And the general solution of the characteristic equation is; And the sequence is periodic with the period: 3 3 -1 = 26 and the sequence is; 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 …. (37) The reciprocal polynomial of the prime f(x) is the prime polynomial and it is the characteristic polynomial of the recurring sequence 0 We can see the properties ;   1  25  3  17  3  3   3  17  3  23  3  2   2  23  3  25  3 Is very easy looking that of f(x) and the general term b n is; And the sequence is periodic with the period: 3 3 -1 = 26 and the sequence is; The general term of the sequence {a n }is of the form; Solving the following equations ; We can see the properties ;   1  22  3  16  3  3   3  16  3  14  3  2   2  14  3  22  3 And the general solution of the characteristic equation is; And the sequence is periodic with the period: 3 3 -1 = 26 and the sequence is; 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 …. (37) The reciprocal polynomial of the prime f(x) is the prime polynomial and it is the characteristic polynomial of the recurring sequence 0 the roots of g(x) are; We can see the properties ;   1  25  3  17  3  3   3  17  3  23  3  2   2  23  3  25  3 Is very easy looking that 17 23 25 , , corresponding the roots We can see the properties; The reciprocal polynomial of the prime f(x) is the prime polynomial g is the characteristic polynomial of the recurring sequence 2 2 We can see the properties ;   1  25  3  17  3  3   3  17  3  23  3  2   2  23  3  25  3 of f(x) and the general term b n is; Is very easy looking that We can see the properties ;   1  25  3  17  3  3   3  17  3  23  3  2   2  23  3  25  3 of f(x) and the general term b n is; are roots of the characteristic polynomial g(x) corresponding the roots And the sequence is periodic with the period: 3 3 -1 = 26 and the sequen 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 …. (37) The reciprocal polynomial of the prime f(x) is the prime polynomial g is the characteristic polynomial of the recurring sequence 2 2 We can see the properties ;   1  25  3  17  3  3   3  17  3  23  3  2   2  23  3  25  3 Is very easy looking that of f(x) and the general term b n is; We can see the properties; And the general term of the sequence {b n }is; The sequence {b n }is periodic with the period 26 and it is th 2 1 1 2 1 0 1 0 0 2 2 2 0 1 2 2 1 2 0 2 0 0 1 1 1 0 2 1 1 2 1 0 1 0 0 …. Figure 6 shows register generating {b n }; We can look that one period of the sequence {b n }is one period of th reading it by inverse from the right to the left.
Suppose the multiplication sequence {z n }, where z n = a n .b n , we hav Thus, the sequence {z n } is a linear nonhomogeneous sequence w homogeneous equivalent is equals 6 that is equal to We can see the properties; And the general term of the sequence {b n }is; The sequence {b n }is periodic with the period 26 and it is the follow 2 1 1 2 1 0 1 0 0 2 2 2 0 1 2 2 1 2 0 2 0 0 1 1 1 0 2 1 1 2 1 0 1 0 0 …. Figure 6 shows register generating {b n }; We can look that one period of the sequence {b n }is one period of the seque reading it by inverse from the right to the left.
Suppose the multiplication sequence {z n }, where z n = a n .b n , we have; Thus, the sequence {z n } is a linear nonhomogeneous sequence with the homogeneous equivalent is equals 6 that is equal to We can see the properties; We can see the properties; And the general term of the sequence {b n }is; The sequence {b n }is periodic with the period 26 and it is th 2 1 1 2 1 0 1 0 0 2 2 2 0 1 2 2 1 2 0 2 0 0 1 1 1 0 2 1 1 2 1 0 1 0 0 …. Figure 6 shows register generating {b n }; We can look that one period of the sequence {b n }is one period of th reading it by inverse from the right to the left.
Suppose the multiplication sequence {z n }, where z n = a n .b n , we hav Thus, the sequence {z n } is a linear nonhomogeneous sequence w homogeneous equivalent is equals 6 that is equal to We can see the properties; And the general term of the sequence {b n }is; The sequence {b n }is periodic with the period 26 and it is the fo 2 1 1 2 1 0 1 0 0 2 2 2 0 1 2 2 1 2 0 2 0 0 1 1 1 0 2 1 1 2 1 0 1 0 0 …. Figure 6 shows register generating {b n }; We can look that one period of the sequence {b n }is one period of the s reading it by inverse from the right to the left.
Suppose the multiplication sequence {z n }, where z n = a n .b n , we have; Thus, the sequence {z n } is a linear nonhomogeneous sequence with homogeneous equivalent is equals 6 that is equal to The sequence {b n }is periodic with the period 3 3 -1=26 and it is the following sequence; 2 1 1 2 1 0 1 0 0 2 2 2 0 1 2 2 1 2 0 2 0 0 1 1 1 0 2 1 1 2 1 0 1 0 0 …. Figure 6 shows register generating {b n }; We can look that one period of the sequence {b n }is one period of the sequence {a n }but through reading it by inverse from the right to the left.
Suppose the multiplication sequence {z n }, where z n = a n .b n , we have; We can see the properties; And the general term of the sequence {b n }is; The sequence {b n }is periodic with the period 26 and it is the follo 2 1 1 2 1 0 1 0 0 2 2 2 0 1 2 2 1 2 0 2 0 0 1 1 1 0 2 1 1 2 1 0 1 0 0 …. Figure 6 shows register generating {b n }; We can look that one period of the sequence {b n }is one period of the sequ reading it by inverse from the right to the left.
Suppose the multiplication sequence {z n }, where z n = a n .b n , we have; Thus, the sequence {z n } is a linear nonhomogeneous sequence with th homogeneous equivalent is equals 6 that is equal to x f x f 13, and the sequence {z n } is: 0 1 1 2 0 0 2 0 0 2 1 1 0 0 1 1 2 0 0 2 0 0 2 1 1 0 … Figure 7 illustrates the linear feedback shift registers generating the seque We can see the properties; The sequence {b n }is periodic with the period 26 1 3 3   and it is the following sequence; 2 1 1 2 1 0 1 0 0 2 2 2 0 1 2 2 1 2 0 2 0 0 1 1 1 0 2 1 1 2 1 0 1 0 0 …. Figure 6 shows register generating {b n }; We can look that one period of the sequence {b n }is one period of the sequence {a n }but through reading it by inverse from the right to the left.
Suppose the multiplication sequence {z n }, where z n = a n .b n , we have; Thus, the sequence {z n } is a linear nonhomogeneous sequence with the length of its linear homogeneous equivalent is equals 6 that is equal to x f x f , The period of {z n }is 13, and the sequence {z n } is: 0 1 1 2 0 0 2 0 0 2 1 1 0 0 1 1 2 0 0 2 0 0 2 1 1 0 ……. Figure 7 illustrates the linear feedback shift registers generating the sequence {z n }; We can see the properties; The sequence {b n }is periodic with the period 26 1 3 3   and it is the follo 2 1 1 2 1 0 1 0 0 2 2 2 0 1 2 2 1 2 0 2 0 0 1 1 1 0 2 1 1 2 1 0 1 0 0 …. Figure 6 shows register generating {b n }; We can look that one period of the sequence {b n }is one period of the sequ reading it by inverse from the right to the left.
Suppose the multiplication sequence {z n }, where z n = a n .b n , we have; Thus, the sequence {z n } is a linear nonhomogeneous sequence with th homogeneous equivalent is equals 6 that is equal to x f x f 13, and the sequence {z n } is: 0 1 1 2 0 0 2 0 0 2 1 1 0 0 1 1 2 0 0 2 0 0 2 1 1 0 … Figure 7 illustrates the linear feedback shift registers generating the seque We can see the properties; The sequence {b n }is periodic with the period 26 1 3 3   and it is the following sequence; 2 1 1 2 1 0 1 0 0 2 2 2 0 1 2 2 1 2 0 2 0 0 1 1 1 0 2 1 1 2 1 0 1 0 0 …. Figure 6 shows register generating {b n }; We can look that one period of the sequence {b n }is one period of the sequence {a n }but through reading it by inverse from the right to the left.
Suppose the multiplication sequence {z n }, where z n = a n .b n , we have; Thus, the sequence {z n } is a linear nonhomogeneous sequence with the length of its linear homogeneous equivalent is equals 6 that is equal to Thus, the sequence {z n } is a linear nonhomogeneous sequence with the length of its linear homogeneous equivalent is equals 6 that is equal to The sequence {b n }is periodic with the period 26 1 3 3   and it is the following s 2 1 1 2 1 0 1 0 0 2 2 2 0 1 2 2 1 2 0 2 0 0 1 1 1 0 2 1 1 2 1 0 1 0 0 …. Figure 6 shows register generating {b n }; We can look that one period of the sequence {b n }is one period of the sequence { reading it by inverse from the right to the left.
Suppose the multiplication sequence {z n }, where z n = a n .b n , we have; Thus, the sequence {z n } is a linear nonhomogeneous sequence with the leng homogeneous equivalent is equals 6 that is equal to x f x f , The p 13, and the sequence {z n } is: 0 1 1 2 0 0 2 0 0 2 1 1 0 0 1 1 2 0 0 2 0 0 2 1 1 0 ……. Figure 7 illustrates the linear feedback shift registers generating the sequence {z n The sequence {b n }is periodic with the period 26 1 3 3   and it is the following sequence; 2 1 1 2 1 0 1 0 0 2 2 2 0 1 2 2 1 2 0 2 0 0 1 1 1 0 2 1 1 2 1 0 1 0 0 …. Figure 6 shows register generating {b n }; We can look that one period of the sequence {b n }is one period of the sequence {a n }but through reading it by inverse from the right to the left.
Suppose the multiplication sequence {z n }, where z n = a n .b n , we have; Thus, the sequence {z n } is a linear nonhomogeneous sequence with the length of its linear homogeneous equivalent is equals 6 that is equal to We can see that; 1 ) )(       We can see that; We can check that the set of the all periodic permutation of one period is not an orthogonal set for example, for one permutation of the period: 0 1 1 2 0 0 2 0 0 2 1 1 0 is: 0 0 1 1 2 0 0 2 0 0 2 1 1and the sum of the two vectors is 0 1 2 0 2 0 2 2 0 2 0 2 1 and is don't belong to the set of all permutations of the first period. The linear homogeneous part of z n is LHP(z n ) ={ ' n z }; The characteristic equation of { ' n z } is; We can see that; And the characteristic equation is of the form; 0 1
P5. The length of the linear homogeneous part of the sequence {z n }is equal to 2 ((deg ( )) deg ( )) f x f x − Where the f(x) is the characteristic polynomial of the sequence {S n }.
P6. The period of the sequence {S n } equals the period of the sequence{ n ζ }and equals (p k -1) but the period of the sequence {z n }in the examples1 to 3 is equal to half of this value but in example 4 is equals the quarter of this value. P 7. The set of all cyclic permutations of one period of the multiplication sequence {z n } is not orthogonal set. P 8. There is important difference between the properties of multiplication M-Sequences and M P -Sequences.  furthermore it the permutations between the "coefficients and roots which are in formula of S n " will not be in the formula of n ζ as in the binary M-Sequences.