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arXiv:1001.0001v1 [cs.IT] 30 Dec 2009On the structure of non-full-rank perfect codes
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Olof Heden and Denis S. Krotov∗
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Abstract
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The Krotov combining construction of perfect 1-error-corr ecting binary codes
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from 2000 and a theorem of Heden saying that every non-full-r ank perfect 1-error-
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correcting binary code can be constructed by this combining construction is gener-
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alized to the q-ary case. Simply, every non-full-rank perfect code Cis the union of
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a well-defined family of ¯ µ-components K¯µ, where ¯µbelongs to an “outer” perfect
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codeC⋆, and these components are at distance three from each other. Compo-
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nents from distinct codes can thus freely be combined to obta in new perfect codes.
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The Phelps general product construction of perfect binary c ode from 1984 is gen-
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eralized to obtain ¯ µ-components, and new lower bounds on the number of perfect
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1-error-correcting q-ary codes are presented.
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1. Introduction
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LetFqdenote the finite field with qelements. A perfect1-error-correcting q-ary code of
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lengthn, for short here a perfect code , is a subset Cof the direct product Fn
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q, ofncopies of
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Fq, having the property that any element of Fn
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qdiffers in at most one coordinate position
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from a unique element of C.
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The family of all perfect codes is far from classified or enumerated. We will in this
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short note say something about the structure of these codes. W e need the concept of
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rank.
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We consider Fn
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qas a vector space of dimension nover the finite field Fq. Therank
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of aq-ary codeC, here denoted rank( C), is the dimension of the linear span < C >of
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the elements of C. Trivial, and well known, counting arguments give that if there exist s
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a perfect code in Fn
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qthenn= (qm−1)/(q−1), for some integer m, and|C|=qn−m. So,
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for every perfect code C,
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n−m≤rank(C)≤n.
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If rank(C) =nwe will say that Chasfull rank.
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∗This research collaboration was partially supported by a grant from Swedish Institute; the work of
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the second author was partially supported by the Federal Target Program “Scientific and Educational
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PersonnelofInnovation Russia”for 2009-2013(governmentco ntract No. 02.740.11.0429)and the Russian
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Foundation for Basic Research (grant 08-01-00673).
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1We will show thatevery non-full-rankperfect code isa unionofso ca lled ¯µ-components
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K¯µ, and that these components may be enumerated by some other pe rfect codeC⋆, i.e,
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¯µ∈C⋆. Further, the distance between any two such components will be a t least three.
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This implies that we will be completely free to combine ¯ µ-components from different
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perfect codesofsamelength, toobtainotherperfect codes. Ge neralizing aconstruction by
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Phelps of perfect 1-error correcting binary codes [8], we will obtain further ¯µ-components.
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As an application of our results we will be able to slightly improve the lowe r bound on
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the number of perfect codes given in [6].
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Our results generalize corresponding results for the binary case. In [3] it was shown
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that a binary perfect code can be constructed as the union of diffe rent subcodes (¯ µ-
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components) satisfying some generalized parity-check property , each of them being con-
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structed independently or taken from another perfect code. In [2] it was shown that every
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non-full-rank perfect binary code can be obtained by this combining construction.
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2. Every non-full-rank perfect code is the union of ¯µ-
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components
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We start with some notation. Assume we have positive integers n,t,n1, ...,ntsuch that
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n1+...+nt≤n. Anyq-aryword ¯xwill berepresented intheblockform ¯ x= (¯x1|¯x2|...|
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¯xt|¯x0) = (¯x∗|¯x0), where ¯xi= (xi1,xi2,...,x ini),i= 0,1,...,t,n0=n−n1−...−nt,
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¯x∗= (¯x1|¯x2|...|¯xt). For every block ¯ xi,i= 1,2,...,t, we define σi(¯xi) by
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σi(¯xi) =ni/summationdisplay
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j=1xij,
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and, for ¯x,
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¯σ(¯x) = ¯σ(¯x∗) = (σ1(¯x1),σ2(¯x2),...,σ t(¯xt))
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Recall that the Hamming distance d(¯x,¯y) between two words ¯ x, ¯yof the same length
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means the number of positions in which they differ.
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Amonomial transformation is a map of the space Fn
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qthat can be composed by a
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permutation of the set of coordinate positions and the multiplication in each coordinate
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position with some non-zero element of the finite field Fq.
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Aq-ary codeCislinearifCis a subspace of Fn
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q. A linear perfect code is called a
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Hamming code .
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Theorem 1. LetCbe any non-full-rank perfect code Cof lengthn= (qm−1)/(q−1).
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To any integer r<m, satisfying
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1≤r≤n−rank(C),
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there is aq-ary Hamming code C⋆of lengtht= (qr−1)/(q−1), such that for some
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monomial transformation ψ
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ψ(C) =/uniondisplay
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¯µ∈C⋆K¯µ,
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2where
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K¯µ={(¯x1|¯x2|...|¯xt|¯x0) : ¯σ(¯x) = ¯µ,¯x1,¯x2,...,¯xt∈Fqs
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q,¯x0∈C¯µ(¯x∗)}(1)
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for some family of perfect codes C¯µ(¯x), of length 1+q+q2+...+qs−1, wheres=m−r,
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and satisfying, for each ¯µ∈C⋆,
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d(¯x∗,¯x′
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∗)≤2 =⇒C¯µ(¯x∗)∩C¯µ(¯x′
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∗) =∅. (2)
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The codeC⋆will be called an outercode toψ(C). The subcodes K¯µwill be called
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¯µ-components ofψ(C). As the minimum distance of Cis three, the distance between any
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two distinct ¯ µ-components will be at least three.
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Proof. LetDbe any subspace of Fn
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qcontaining<C >, and of dimension n−r. By
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using a monomial transformation ψof space we may achieve that the dual space of ψ(D)
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is the nullspace of a r×n-matrix
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H=
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| | | | | | | |
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¯α11···¯α1n1¯α21···¯α2n2···¯αt1···¯αtnt¯0···¯0
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| | | | | | | |
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where ¯αij= ¯αi, fori= 1,2,...,t, the first non-zero coordinate in each vector ¯ αiequals
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1, ¯αi/ne}ationslash= ¯αi′, fori/ne}ationslash=i′, and where the columns of Hare in lexicographic order, according
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to some given ordering of Fq.
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To avoid too much notation we assume that Cwas such that ψ= id.
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LetC⋆be the null space of the matrix
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