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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

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