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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