1001.0001.txt

arXiv:1001.0001v1  [cs.IT]  30 Dec 2009On the structure of non-full-rank perfect codes
Olof Heden and Denis S. Krotov∗
Abstract
The Krotov combining construction of perfect 1-error-corr ecting binary codes
from 2000 and a theorem of Heden saying that every non-full-r ank perfect 1-error-
correcting binary code can be constructed by this combining construction is gener-
alized to the q-ary case. Simply, every non-full-rank perfect code Cis the union of
a well-defined family of ¯ µ-components K¯µ, where ¯µbelongs to an “outer” perfect
codeC⋆, and these components are at distance three from each other. Compo-
nents from distinct codes can thus freely be combined to obta in new perfect codes.
The Phelps general product construction of perfect binary c ode from 1984 is gen-
eralized to obtain ¯ µ-components, and new lower bounds on the number of perfect
1-error-correcting q-ary codes are presented.
1. Introduction
LetFqdenote the finite field with qelements. A perfect1-error-correcting q-ary code of
lengthn, for short here a perfect code , is a subset Cof the direct product Fn
q, ofncopies of
Fq, having the property that any element of Fn
qdiffers in at most one coordinate position
from a unique element of C.
The family of all perfect codes is far from classified or enumerated. We will in this
short note say something about the structure of these codes. W e need the concept of
rank.
We consider Fn
qas a vector space of dimension nover the finite field Fq. Therank
of aq-ary codeC, here denoted rank( C), is the dimension of the linear span < C >of
the elements of C. Trivial, and well known, counting arguments give that if there exist s
a perfect code in Fn
qthenn= (qm−1)/(q−1), for some integer m, and|C|=qn−m. So,
for every perfect code C,
n−m≤rank(C)≤n.
If rank(C) =nwe will say that Chasfull rank.
∗This research collaboration was partially supported by a grant from Swedish Institute; the work of
the second author was partially supported by the Federal Target Program “Scientific and Educational
PersonnelofInnovation Russia”for 2009-2013(governmentco ntract No. 02.740.11.0429)and the Russian
Foundation for Basic Research (grant 08-01-00673).
1We will show thatevery non-full-rankperfect code isa unionofso ca lled ¯µ-components
K¯µ, and that these components may be enumerated by some other pe rfect codeC⋆, i.e,
¯µ∈C⋆. Further, the distance between any two such components will be a t least three.
This implies that we will be completely free to combine ¯ µ-components from different
perfect codesofsamelength, toobtainotherperfect codes. Ge neralizing aconstruction by
Phelps of perfect 1-error correcting binary codes [8], we will obtain further ¯µ-components.
As an application of our results we will be able to slightly improve the lowe r bound on
the number of perfect codes given in [6].
Our results generalize corresponding results for the binary case. In [3] it was shown
that a binary perfect code can be constructed as the union of diffe rent subcodes (¯ µ-
components) satisfying some generalized parity-check property , each of them being con-
structed independently or taken from another perfect code. In [2] it was shown that every
non-full-rank perfect binary code can be obtained by this combining construction.
2. Every non-full-rank perfect code is the union of ¯µ-
components
We start with some notation. Assume we have positive integers n,t,n1, ...,ntsuch that
n1+...+nt≤n. Anyq-aryword ¯xwill berepresented intheblockform ¯ x= (¯x1|¯x2|...|
¯xt|¯x0) = (¯x∗|¯x0), where ¯xi= (xi1,xi2,...,x ini),i= 0,1,...,t,n0=n−n1−...−nt,
¯x∗= (¯x1|¯x2|...|¯xt). For every block ¯ xi,i= 1,2,...,t, we define σi(¯xi) by
σi(¯xi) =ni/summationdisplay
j=1xij,
and, for ¯x,
¯σ(¯x) = ¯σ(¯x∗) = (σ1(¯x1),σ2(¯x2),...,σ t(¯xt))
Recall that the Hamming distance d(¯x,¯y) between two words ¯ x, ¯yof the same length
means the number of positions in which they differ.
Amonomial transformation is a map of the space Fn
qthat can be composed by a
permutation of the set of coordinate positions and the multiplication in each coordinate
position with some non-zero element of the finite field Fq.
Aq-ary codeCislinearifCis a subspace of Fn
q. A linear perfect code is called a
Hamming code .
Theorem 1. LetCbe any non-full-rank perfect code Cof lengthn= (qm−1)/(q−1).
To any integer r<m, satisfying
1≤r≤n−rank(C),
there is aq-ary Hamming code C⋆of lengtht= (qr−1)/(q−1), such that for some
monomial transformation ψ
ψ(C) =/uniondisplay
¯µ∈C⋆K¯µ,
2where
K¯µ={(¯x1|¯x2|...|¯xt|¯x0) : ¯σ(¯x) = ¯µ,¯x1,¯x2,...,¯xt∈Fqs
q,¯x0∈C¯µ(¯x∗)}(1)
for some family of perfect codes C¯µ(¯x), of length 1+q+q2+...+qs−1, wheres=m−r,
and satisfying, for each ¯µ∈C⋆,
d(¯x∗,¯x′
∗)≤2 =⇒C¯µ(¯x∗)∩C¯µ(¯x′
∗) =∅. (2)
The codeC⋆will be called an outercode toψ(C). The subcodes K¯µwill be called
¯µ-components ofψ(C). As the minimum distance of Cis three, the distance between any
two distinct ¯ µ-components will be at least three.
Proof. LetDbe any subspace of Fn
qcontaining<C >, and of dimension n−r. By
using a monomial transformation ψof space we may achieve that the dual space of ψ(D)
is the nullspace of a r×n-matrix
H=
| | | | | | | |
¯α11···¯α1n1¯α21···¯α2n2···¯αt1···¯αtnt¯0···¯0
| | | | | | | |

where ¯αij= ¯αi, fori= 1,2,...,t, the first non-zero coordinate in each vector ¯ αiequals
1, ¯αi/ne}ationslash= ¯αi′, fori/ne}ationslash=i′, and where the columns of Hare in lexicographic order, according
to some given ordering of Fq.
To avoid too much notation we assume that Cwas such that ψ= id.
LetC⋆be the null space of the matrix
H⋆=
| | |
¯α1¯α2···¯αt
| | |

Define, for ¯ µ∈C⋆,
K¯µ={(¯x1|¯x2|...|¯xt|¯x0)∈C: (σ1(¯x1),σ2(¯x2),...,σ(¯xt)) = ¯µ}.
Then,
C=/uniondisplay
¯µ∈C⋆K¯µ.
Further, since any two columns of H⋆are linearly independent, for any two distinct words
¯µand ¯µ′ofC⋆
d(K¯µ,K¯µ′)≥3. (3)
We will show that K¯µhas the properties given in Equation (1).
Any word ¯x= (¯x1|¯x2|...|¯xt|¯x0) must be at distance at most one from a word of
C, and hence, the word ( σ1(¯x1),σ2(¯x2),...,σ t(¯xt)) is at distance at most one from some
word ofC⋆. It follows that C⋆is a perfect code, and as a consequence, as C⋆is linear, it
is a Hamming code with parity-check matrix H⋆. As the number of rows of H⋆isr, we
then get that the number tof columns of H⋆is equal to
t=qr−1
q−1= 1+q+q2+...+qr−1.
3For any word ¯ x∗ofFn1+n2+...+ntq with ¯σ(¯x∗) = ¯µ∈C⋆, we now define the code C¯µ(¯x∗)
of lengthn0by
C¯µ(¯x∗) ={¯c∈Fn0
q: (¯x∗|¯c)∈C}.
Again, using the fact that Cis a perfect code, we may deduce that for any ¯ x∗such
that the set C¯µ(¯x∗) is non empty, the set C¯µ(¯x∗) must be a perfect code of length n0=
(qs−1)/(q−1), for some integer s.
From the fact that the minimum distance of Cequals three, we get the property in
Equation (2).
Let ¯eidenote a word of weight one with the entry 1 in the coordinate positio ni. It
then follows that the two perfect codes C¯µ(¯x∗) andC¯µ(¯x∗+ ¯e1−¯ei), fori= 2,3,...,n 1,
must be mutually disjoint. Hence, n1is at most equal to the number of perfect codes in
a partition of Fn0qinto perfect codes, i.e.,
n1≤(q−1)n0+1 =qs.
Similarly,ni≤qs, fori= 2,3,...,t.
Reversing these arguments, using Equation (3) and the fact that Cis a perfect code,
we find that ni, for eachi= 1,2,...,t, is at least equal to the number of words in an
1-ball ofFn0q.
We conclude that ni=qs, fori= 1,2,...,t, and finally
n=qs(1+q+q2+...+qr−1)+1+q+q2+...+qs−1= 1+q+q2+...+qr+s−1.
Givenr, we can then find sfrom the equality
n= 1+q+q2+...+qm−1.
△
3. Combining construction of perfect codes
In the previous section, it was shown that a perfect code, depend ing on its rank, can
be divided onto small or large number of so-called ¯ µ-components, which satisfy some
equation with ¯ σ. The construction described in the following theorem realizes the ide a
of combining independent ¯ µ-components, differently constructed or taken from different
perfect codes, in one perfect code.
A functionf: Σn→Σ, where Σ is some set, is called an n-ary(ormultary)quasigroup
of order |Σ|if in the equality z0=f(z1,...,z n) knowledge of any nelements of z0,z1,
...,znuniquely specifies the remaining one.
Theorem 2. Letmandrbe integers, m>r,qbe a prime power, n= (qm−1)/(q−1)
andt= (qr−1)/(q−1). Assume that C∗is a perfect code in Ft
qand for every ¯µ∈C∗
we have a distance- 3codeK¯µ⊂Fn
qof cardinality qn−m−(t−r)that satisfies the following
generalized parity-check law:
¯σ(¯x) = (σ1(x1,...,x l),...,σ t(xlt−l+1,...,x lt)) = ¯µ
4for every ¯x= (x1,...,x n)∈K¯µ, wherel=qm−rand¯σ= (σ1,...,σ t)is a collections of
l-ary quasigroups of order q. Then the union
C=/uniondisplay
¯µ∈C∗K¯µ
is a perfect code in Fn
q.
Proof. It is easy to check that Chas the cardinality of a perfect code. The distance
at least 3 between different words ¯ x, ¯yfromCfollows from the code distances of K¯µ(if
¯x, ¯ybelong to the same K¯µ) andC∗(if ¯x, ¯ybelong to different K¯µ′,K¯µ′′, ¯µ′,¯µ′′∈C∗).△
The ¯µ-components K¯µcanbeconstructedindependentlyortakenfromdifferentperfec t
codes. In the important case when all σiare linear quasigroups (e.g., σi(y1,...,y l) =
y1+...+yl) the components can be taken from any perfect code of rank at m ostn−r, as
followsfromtheprevioussection(itshouldbenotedthatif ¯ σislinear, thena ¯ µ-component
can be obtained from any ¯ µ′-component by adding a vector ¯ zsuch that ¯σ(¯z) = ¯µ−¯µ′).
In general, the existence of ¯ µ-components that satisfy the generalized parity-check law
for arbitrary ¯ σis questionable. But for some class of ¯ σsuch components exist, as we will
see from the following two subsections.
Remark. It is worth mentioning that ¯ µ-components can exist for arbitrary length tof
¯µ(for example, in the next two subsections there are no restriction s ont), if we do not
require the possibility to combine them into a perfect code. This is esp ecially important
for the study of perfect codes of small ranks (close to the rank o f a linear perfect code):
once we realize that the code is the union of ¯ µ-components of some special form, we may
forget about the code length and consider ¯ µ-components for arbitrary length of ¯ µ, which
allows to use recursive approaches.
3.1. Mollard-Phelps construction
Here we describe the way to construct ¯ µ-components derived from the product construc-
tion discovered independently in [7] and [9]. In terms of ¯ µ-components, the construction
in [9] is more general; it allows substitution of arbitrary multary quasig roups, and we will
use this possibility in Section 4.
Lemma 1. Let¯µ∈Ft
qand letC#be a perfect code in Fk
q. Letvandhbe(q−1)-ary
quasigroups of order qsuch that the code {(¯y|v(¯y)|h(¯y)) : ¯y∈Fq−1
q}is perfect. Let
V1, ...,VtandH1, ...,Hkbe respectively (k+1)-ary and (t+1)-ary quasigroups of order
q. Then the set
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,
(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.
8. K. T. Phelps. A general product construction for error corre cting codes. SIAM J.
Algebraic Discrete Methods , 5(2):224–228, 1984.
9. K. T. Phelps. A product construction for perfect codes over a rbitrary alphabets.
IEEE Trans. Inf. Theory , 30(5):769–771, 1984.
10. V. N. Potapov and D. S. Krotov. On the number of n-ary quasigroups of finite order.
Submitted. ArXiv:0912.5453
11. V.N.PotapovandD.S.Krotov. Asymptoticsforthenumbero fn-quasigroupsoforder
4.Sib. Math. J. , 47(4):720–731, 2006. DOI: 10.1007/s11202-006-0083-9 tran slated
from Sib. Mat. Zh. 47(4) (2006), 873-887. ArXiv:math/0605104
O. Heden
Department of Mathematics, KTH
S-100 44 Stockholm, Sweden
email:olohed@math.kth.se
D. Krotov
Sobolev Institute of Mathematics
and
Mechanics and Mathematics Department, Novosibirsk State Univer sity
Novosibirsk, Russia
email:krotov@math.nsc.ru
8