On the classification of perfect codes: side class structures

The side class structure of a perfect 1-error correcting binary code (hereafter referred to as a perfect code) C describes the linear relations between the coset representatives of the kernel of C. Two perfect codes C and C′ are linearly equivalent if there exists a non-singular matrix A such that AC = C′ where C and C′ are matrices with the code words of C and C′ as columns. Hessler proved that the perfect codes C and C′ are linearly equivalent if and only if they have isomorphic side class structures. The aim of this paper is to describe all side class structures. It is shown that the transpose of any side class structure is the dual of a subspace of the kernel of some perfect code and vice versa; any dual of a subspace of a kernel of some perfect code is the transpose of the side class structure of some perfect code. The conclusion is that for classification purposes of perfect codes it is sufficient to find the family of all kernels of perfect codes.     

On diameter perfect constant-weight ternary codes

From cosets of binary Hamming codes we construct diameter perfect constant-weight ternary codes with weight n-1 (where n is the code length) and distances 3 and 5. The class of distance 5 codes has parameters unknown before.     

On perfect binary codes

Some results on perfect codes obtained during the last 6 years are discussed. The main methods to construct perfect codes such as the method of -components and the concatenation approach and their implementations to solve some important problems are analyzed. The solution of the ranks and kernels problem, the lower and upper bounds of the automorphism group order of a perfect code, spectral properties, diameter perfect codes, isometries of perfect codes and codes close to them by close-packed properties are considered.     

The non-existence of some perfect codes over non-prime power alphabets

Let denote the number of times the prime number p appears in the prime factorization of the integer q. The following result is proved: If there is a perfect 1-error correcting code of length n over an alphabet with q symbols then, for every prime number .This condition is stronger than both the packing condition and the necessary condition given by the Lloyd theorem, as it for example excludes the existence of a perfect code with the parameters (n,q,e)=(19,6,1).     

Maximal partial packings of $${Z_2^n}$$ with perfect codes

A maximal partial Hamming packing of is a family of mutually disjoint translates of Hamming codes of length n, such that any translate of any Hamming code of length n intersects at least one of the translates of Hamming codes in . The number of translates of Hamming codes in is the packing number, and a partial Hamming packing is strictly partial if the family does not constitute a partition of . A simple and useful condition describing when two translates of Hamming codes are disjoint or not disjoint is proved. This condition depends on the dual codes of the corresponding Hamming codes. Partly, by using this condition, it is shown that the packing number p, for any maximal strictly partial Hamming packing of , n = 2 ^m −1, satisfies . It is also proved that for any n equal to 2 ^m −1, , there exist maximal strictly partial Hamming packings of with packing numbers n−10,n−9,n−8,...,n−1. This implies that the upper bound is tight for any n = 2 ^m −1, . All packing numbers for maximal strictly partial Hamming packings of , n = 7 and 15, are found by a computer search. In the case n = 7 the packing number is 5, and in the case n = 15 the possible packing numbers are 5,6,7,...,13 and 14.      

A remark on full rank perfect codes

Any full rank perfect 1-error correcting binary code of length n=2^k-1 and with a kernel of dimension n-log(n+1)-m, where m is sufficiently large, may be used to construct a full rank perfect 1-error correcting binary code of length 2^m-1 and with a kernel of dimension n-log(n+1)-k. Especially we may construct full rank perfect 1-error correcting binary codes of length n=2^m-1 and with a kernel of dimension n-log(n+1)-4 for m=6,7,…,10.This result extends known results on the possibilities for the size of a kernel of a full rank perfect code.     

On perfect Lee codes

In this paper we survey recent results on the Golomb-Welch conjecture and its generalizations and variations. We also show that there are no perfect 2-error correcting Lee codes of block length 5 and 6 over Z. This provides additional support for the Golomb Welch conjecture as it settles the two smallest cases open so far.     

On kernels of perfect codes

It is shown that there exists a perfect one-error-correcting binary code with a kernel which is not contained in any Hamming code.     

Generalized MacWilliams identities and their applications to perfect binary codes

We present generalized MacWilliams identities for binary codes. These identities naturally lead to the concepts of the local weight distribution of a binary code with respect to a word u and its MacWilliams u-transform. In the case that u is the all-one word, these ones correspond to the weight distribution of a binary code and its MacWilliams transform, respectively. We identify a word v with its support, and consider v as a subset of {1, 2,..., n}. For two words u,w of length n such that their intersection is the empty set, define the u-face centered at w to be the set . A connection between our MacWilliams u-transform and the weight distribution of a binary code in the u-face centered at the zero word is presented. As their applications, we also investigate the properties of a perfect binary code. For a perfect binary code C, the main results are as follows: first, it is proved that our local weight distribution of C is uniquely determined by the number of codewords of C in the orthogonal u-face centered at the zero word. Next, we give a direct proof for the known result, concerning the weight distribution of a coset of C in the u-face centered at the zero word, by A. Y. Vasil’eva without using induction. Finally, it is proved that the weight distribution of C in the orthogonal u-face centered at w is uniquely determined by the codewords of C in the u-face centered at the zero word.      

On the size of the symmetry group of a perfect code

It is shown that for every nonlinear perfect code C of length n and rank r with n−log(n+1)+1≤r≤n−1, where denotes the group of symmetries of C. This bound considerably improves a bound of Malyugin.     

Perfect codes from the dual point of view I

The two concepts dual code and parity check matrix for a linear perfect 1-error correcting binary code are generalized to the case of non-linear perfect codes. We show how this generalization can be used to enumerate some particular classes of perfect 1-error correcting binary codes. We also use it to give an answer to a problem of Avgustinovich: whether or not the kernel of every perfect 1-error correcting binary code is always contained in some Hamming code.     

On perfect <Emphasis Type="Italic">p</Emphasis>-ary codes of length <Emphasis Type="Italic">p</Emphasis> + 1

Let p be a prime number and assume p ≥ 5. We will use a result of L. Redéi to prove, that every perfect 1-error correcting code C of length p + 1 over an alphabet of cardinality p, such that C has a rank equal to p and a kernel of dimension p − 2, will be equivalent to some Hamming code H. Further, C can be obtained from H, by the permutation of the symbols, in just one coordinate position.      

Intersections of q-ary perfect codes

The intersections of q-ary perfect codes are under study. We prove that there exist two q-ary perfect codes C ₁ and C ₂ of length N = qn + 1 such that |C ₁ ? C ₂| = k · |P _i |/p for each k ∈ {0,..., p · K ? 2, p · K}, where q = p ^r , p is prime, r ≥ 1, $n = \tfrac{{q^{m - 1} - 1}}{{q - 1}}$ , m ≥ 2, |P _i | = p ^nr(q?2)+n , and K = p ^{n(2r?1)?r(m?1)}. We show also that there exist two q-ary perfect codes of length N which are intersected by p ^nr(q?3)+n codewords.     

On perfect deletion‐correcting codes

In this article, we present constructions for perfect deletion‐correcting codes. The first construction uses perfect deletion‐correcting codes without repetition of letters to construct other perfect deletion‐correcting codes. This is a generalization of the construction shown in 1 . In the third section, we investigate several constructions of perfect deletion‐correcting codes using designs. In the last section, we investigate perfect deletion‐correcting codes containing few codewords. © 2003 Wiley Periodicals, Inc.     

Perfect codes in direct products of cycles—a complete characterization

Let be a direct product of cycles. It is known that for any r1, and any n2, each connected component of G contains a so-called canonical r-perfect code provided that each ℓ_i is a multiple of rⁿ+(r+1)ⁿ. Here we prove that up to a reasonably defined equivalence, these are the only perfect codes that exist.     

On the complexity of SNP block partitioning under the perfect phylogeny model

Recent technologies for typing single nucleotide polymorphisms (SNPs) across a population are producing genome-wide genotype data for tens of thousands of SNP sites. The emergence of such large data sets underscores the importance of algorithms for large-scale haplotyping. Common haplotyping approaches first partition the SNPs into blocks of high linkage-disequilibrium, and then infer haplotypes for each block separately. We investigate an integrated haplotyping approach where a partition of the SNPs into a minimum number of non-contiguous subsets is sought, such that each subset can be haplotyped under the perfect phylogeny model. We show that finding an optimum partition is -hard even if we are guaranteed that two subsets suffice. On the positive side, we show that a variant of the problem, in which each subset is required to admit a perfect path phylogeny haplotyping, is solvable in polynomial time.     

An explicit class of codes with good parameters and their duals

We study a class of codes with good parameters and their duals explicitly. We give direct constructions of the dual codes and obtain self-orthogonal codes with good parameters.