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(V1(v(¯x11),...,v(¯x1k),y1),...,V t(v(¯xt1),...,v(¯xtk),yt)) = ¯µ,
(H1(h(¯x11),...,h(¯xt1),z1),...,H k(h(¯x1k),...,h(¯xtk),zk))∈C#/bracerightBig
is a¯µ-component that satisﬁes the generalized parity-check law with
σi(·,...,·,·) =Vi(v(·),...,v(·),·).
5(The elements of F(q−1)kt+k+t
q in this construction may be thought of as three-dimensional
arrays where the elements of ¯xijare z-lined, every underlined block is y-lined, and the
tuple of blocks is x-lined. Naturally, the multary quasigroups Vimay be named “vertical”
andHi, “horizontal”.)
The proof of the code distance is similar to that in [9], and the other pr operties of a
¯µ-component are straightforward. The existence of admissible ( q−1)-ary quasigroups v
andhis the only restriction on the q(this concerns the next subsection as well). If Fqis
a ﬁnite ﬁeld, there are linear examples: v(y1,...,y q−1) =y1+...+yq−1,v(y1,...,y q−1) =
α1y1+...+αq−1yq−1whereα1, ...,αq−1are all the non-zero elements of Fq. Ifqis not
a prime power, the existence of a q-ary perfect code of length q+1 is an open problem
(with the only exception q= 6, when the nonexistence follows from the nonexistence of
two orthogonal 6 ×6 Latin squares [1, Th.6]).
3.2. Generalized Phelps construction
Here we describe another way to construct ¯ µ-components, which generalizes the construc-
tion of binary perfect codes from [8].
Lemma 2. Let¯µ∈Ft
q. Let for every ifrom1tot+1the codesCi,j,j= 0,1,...,qk−k
form a partition of Fk
qinto perfect codes and γi:Fk
q→ {0,1,...,qk−k}be the corre-
sponding partition function:
γi(¯y) =j⇐⇒¯y∈Ci,j.
Letvandhbe(q−1)-ary quasigroups of order qsuch that the code {(¯y\|v(¯y)\|h(¯y)) :
¯y∈Fq−1
q}is perfect. Let V1, ...,Vtbe(k+ 1)-ary quasigroups of order qandQbe a
t-ary quasigroup of order qk−k+1.
K¯µ=/braceleftBig
(¯x11\|...\|¯x1k\|y1\|¯x21\|...\|¯x2k\|y2\|...\|¯xt1\|...\|¯xtk\|yt\|z1\|z2\|...\|zk) :
¯xij∈Fq−1
q,
(V1(v(¯x11),...,v(¯x1k),y1),...,V t(v(¯xt1),...,v(¯xtk),yt)) = ¯µ,
Q(γ1(h(¯x11),...,h(¯x1k)),...,γ t(h(¯xt1),...,h(¯xtk))) =γt+1(z1,...,zk)/bracerightBig
is a¯µ-component that satisﬁes the generalized parity-check law with
σi(·,...,·,·) =Vi(v(·),...,v(·),·).
The proof consists of trivial veriﬁcations.
4. On the number of perfect codes
In this section we discuss some observations, which result in the bes t known lower bound
on the number of q-ary perfect codes, q≥3. The basic facts are already contained in
other known results: lower bounds on the number of multary quasig roups of order q, the
6construction [9] of perfect codes from multary quasigroups of or derq, and the possibility
to choose the quasigroup independently for every vector of the o uter code (this possibility
was not explicitly mentioned in [9], but used in the previous paper [8]).
A general lower bound, in terms of the number of multary quasigrou ps, is given by
Lemma 3. In combination with Lemma 4, it gives explicit numbers.
Lemma 3. The number of q-ary perfect codes of length nis not less than
Q/parenleftBiggn−1
q,q/parenrightBiggRn−1
q
whereQ(m,q)is the number of m-ary quasigroups of order qand whereRn′=qn′/(n′q−
q+1)is the cardinality of a perfect code of length n′.
Proof. Constructing a perfect code like in Theorem 2 with t=n−1
q, we combine
Rn−1
qdiﬀerent ¯µ-components.
Constructing every such a component as in Lemma 2, k= 1,t=n−1
q, we are free
to choose the t-ary quasigroup Qof orderqinQ(t,q) ways. Clearly, diﬀerent t-ary
quasigroups give diﬀerent components. (Equivalently, we can use L emma 1 and choose
the (t+1)-ary quasigroup H1, but should note that the value of H1in the construction is
always ﬁxed when k= 1, because C#consists of only one vertex; so we again have Q(t,q)
diﬀerent choices, not Q(t+1,q)). △
Lemma 4. The number Q(m,q)ofm-ary quasigroups of order qsatisﬁes:
(a) [5]Q(m,3) = 3·2m;
(b) [11]Q(m,4) = 3m+1·22m+1(1+o(1));
(c) [4]Q(m,5)≥23n/3−0.072;
(d) [10]Q(m,q)≥2((q2−4q+3)/4)n/2for oddq(the previous bound [4]wasQ(m,q)≥
2⌊q/3⌋n);
(e) [4]Q(m,q1q2)≥Q(m,q1)·Q(m,q2)qm
1.
For oddq≥5, the number of codes given by Lemmas 3 and 4(c,d) improves the
constantcin the lower estimation of form eecn(1+o(1))for the number of perfect codes, in
comparison with the last known lower bound [6]. Informally, this can be explained in the
following way: the construction in [6] can be described in terms of mu tually independent
small modiﬁcations of the linear multary quasigroup of order q, while the lower bounds
in Lemma 4(c,d) are based on a specially-constructed nonlinear multa ry quasigroup that
allows a lager number of independent modiﬁcations. For q= 3 andq= 2s, the number
of codes given by Lemmas 3 and 4(a,b,e) also slightly improves the boun d in [6], but do
not aﬀect on the constant c.
7References
1. S. W. Golomb and E. C. Posner. Rook domains, latin squares, and e rror-distributing
codes.IEEE Trans. Inf. Theory , 10(3):196–208, 1964.
2. O. Heden. On the classiﬁcation of perfect binary 1-error corre cting codes. Preprint
TRITA-MAT-2002-01, KTH, Stockholm, 2002.
3. D. S. Krotov. Combining construction of perfect binary codes. Probl. Inf. Transm. ,
36(4):349–353, 2000. translated from Probl. Peredachi Inf. 36 (4) (2000), 74-79.
4. D. S. Krotov, V. N. Potapov, and P. V. Sokolova. On reconstru cting reducible n-ary
quasigroups and switching subquasigroups. Quasigroups Relat. Syst. , 16(1):55–67,
2008. ArXiv:math/0608269
5. C. F. Laywine and G. L. Mullen. Discrete Mathematics Using Latin Squares . Wiley,
New York, 1998.
6. A. V. Los’. Construction of perfect q-ary codes by switchings o f simple components.
Probl. Inf. Transm. , 42(1):30–37, 2006. DOI: 10.1134/S0032946006010030 transla ted
from Probl. Peredachi Inf. 42(1) (2006), 34-42.
7. M. Mollard. A generalized parity function and its use in the constru ction of perfect
codes.SIAM J. Algebraic Discrete Methods , 7(1):113–115, 1986.

(V1(v(¯x11),...,v(¯x1k),y1),...,V t(v(¯xt1),...,v(¯xtk),yt)) = ¯µ,

(H1(h(¯x11),...,h(¯xt1),z1),...,H k(h(¯x1k),...,h(¯xtk),zk))∈C#/bracerightBig

is a¯µ-component that satisﬁes the generalized parity-check law with

σi(·,...,·,·) =Vi(v(·),...,v(·),·).

5(The elements of F(q−1)kt+k+t

q in this construction may be thought of as three-dimensional

arrays where the elements of ¯xijare z-lined, every underlined block is y-lined, and the

tuple of blocks is x-lined. Naturally, the multary quasigroups Vimay be named “vertical”

andHi, “horizontal”.)

The proof of the code distance is similar to that in [9], and the other pr operties of a

¯µ-component are straightforward. The existence of admissible ( q−1)-ary quasigroups v

andhis the only restriction on the q(this concerns the next subsection as well). If Fqis

a ﬁnite ﬁeld, there are linear examples: v(y1,...,y q−1) =y1+...+yq−1,v(y1,...,y q−1) =

α1y1+...+αq−1yq−1whereα1, ...,αq−1are all the non-zero elements of Fq. Ifqis not

a prime power, the existence of a q-ary perfect code of length q+1 is an open problem

(with the only exception q= 6, when the nonexistence follows from the nonexistence of

two orthogonal 6 ×6 Latin squares [1, Th.6]).

3.2. Generalized Phelps construction

Here we describe another way to construct ¯ µ-components, which generalizes the construc-

tion of binary perfect codes from [8].

Lemma 2. Let¯µ∈Ft

q. Let for every ifrom1tot+1the codesCi,j,j= 0,1,...,qk−k

form a partition of Fk

qinto perfect codes and γi:Fk

q→ {0,1,...,qk−k}be the corre-

sponding partition function:

γi(¯y) =j⇐⇒¯y∈Ci,j.

Letvandhbe(q−1)-ary quasigroups of order qsuch that the code {(¯y|v(¯y)|h(¯y)) :

¯y∈Fq−1

q}is perfect. Let V1, ...,Vtbe(k+ 1)-ary quasigroups of order qandQbe a

t-ary quasigroup of order qk−k+1.

K¯µ=/braceleftBig

(¯x11|...|¯x1k|y1|¯x21|...|¯x2k|y2|...|¯xt1|...|¯xtk|yt|z1|z2|...|zk) :

¯xij∈Fq−1

q,

(V1(v(¯x11),...,v(¯x1k),y1),...,V t(v(¯xt1),...,v(¯xtk),yt)) = ¯µ,

Q(γ1(h(¯x11),...,h(¯x1k)),...,γ t(h(¯xt1),...,h(¯xtk))) =γt+1(z1,...,zk)/bracerightBig

is a¯µ-component that satisﬁes the generalized parity-check law with

σi(·,...,·,·) =Vi(v(·),...,v(·),·).

The proof consists of trivial veriﬁcations.

4. On the number of perfect codes

In this section we discuss some observations, which result in the bes t known lower bound

on the number of q-ary perfect codes, q≥3. The basic facts are already contained in

other known results: lower bounds on the number of multary quasig roups of order q, the

6construction [9] of perfect codes from multary quasigroups of or derq, and the possibility

to choose the quasigroup independently for every vector of the o uter code (this possibility

was not explicitly mentioned in [9], but used in the previous paper [8]).

A general lower bound, in terms of the number of multary quasigrou ps, is given by

Lemma 3. In combination with Lemma 4, it gives explicit numbers.

Lemma 3. The number of q-ary perfect codes of length nis not less than

Q/parenleftBiggn−1

q,q/parenrightBiggRn−1

q

whereQ(m,q)is the number of m-ary quasigroups of order qand whereRn′=qn′/(n′q−

q+1)is the cardinality of a perfect code of length n′.

Proof. Constructing a perfect code like in Theorem 2 with t=n−1

q, we combine

Rn−1

qdiﬀerent ¯µ-components.

Constructing every such a component as in Lemma 2, k= 1,t=n−1

q, we are free

to choose the t-ary quasigroup Qof orderqinQ(t,q) ways. Clearly, diﬀerent t-ary

quasigroups give diﬀerent components. (Equivalently, we can use L emma 1 and choose

the (t+1)-ary quasigroup H1, but should note that the value of H1in the construction is

always ﬁxed when k= 1, because C#consists of only one vertex; so we again have Q(t,q)

diﬀerent choices, not Q(t+1,q)). △

Lemma 4. The number Q(m,q)ofm-ary quasigroups of order qsatisﬁes:

(a) [5]Q(m,3) = 3·2m;

(b) [11]Q(m,4) = 3m+1·22m+1(1+o(1));

(c) [4]Q(m,5)≥23n/3−0.072;

(d) [10]Q(m,q)≥2((q2−4q+3)/4)n/2for oddq(the previous bound [4]wasQ(m,q)≥

2⌊q/3⌋n);

(e) [4]Q(m,q1q2)≥Q(m,q1)·Q(m,q2)qm

1.

For oddq≥5, the number of codes given by Lemmas 3 and 4(c,d) improves the

constantcin the lower estimation of form eecn(1+o(1))for the number of perfect codes, in

comparison with the last known lower bound [6]. Informally, this can be explained in the

following way: the construction in [6] can be described in terms of mu tually independent

small modiﬁcations of the linear multary quasigroup of order q, while the lower bounds

in Lemma 4(c,d) are based on a specially-constructed nonlinear multa ry quasigroup that

allows a lager number of independent modiﬁcations. For q= 3 andq= 2s, the number

of codes given by Lemmas 3 and 4(a,b,e) also slightly improves the boun d in [6], but do

not aﬀect on the constant c.

7References

1. S. W. Golomb and E. C. Posner. Rook domains, latin squares, and e rror-distributing

codes.IEEE Trans. Inf. Theory , 10(3):196–208, 1964.

2. O. Heden. On the classiﬁcation of perfect binary 1-error corre cting codes. Preprint

TRITA-MAT-2002-01, KTH, Stockholm, 2002.

3. D. S. Krotov. Combining construction of perfect binary codes. Probl. Inf. Transm. ,

36(4):349–353, 2000. translated from Probl. Peredachi Inf. 36 (4) (2000), 74-79.

4. D. S. Krotov, V. N. Potapov, and P. V. Sokolova. On reconstru cting reducible n-ary

quasigroups and switching subquasigroups. Quasigroups Relat. Syst. , 16(1):55–67,

2008. ArXiv:math/0608269

5. C. F. Laywine and G. L. Mullen. Discrete Mathematics Using Latin Squares . Wiley,

New York, 1998.

6. A. V. Los’. Construction of perfect q-ary codes by switchings o f simple components.

Probl. Inf. Transm. , 42(1):30–37, 2006. DOI: 10.1134/S0032946006010030 transla ted

from Probl. Peredachi Inf. 42(1) (2006), 34-42.

7. M. Mollard. A generalized parity function and its use in the constru ction of perfect

codes.SIAM J. Algebraic Discrete Methods , 7(1):113–115, 1986.