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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 |
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