Binary extended perfect codes of length 16 and rank 14

All extended binary perfect (16, 4, 2¹¹) codes of rank 14 over the field F ₂ are classified. It is proved that among all nonequivalent extended binary perfect (16, 4, 2¹¹) codes there are exactly 1719 nonequivalent codes of rank 14 over F ₂. Among these codes there are 844 codes classified by Phelps (Solov’eva-Phelps codes) and 875 other codes obtained by the construction of Etzion-Vardy and by a new general doubling construction, presented in the paper. Thus, the only open question in the classification of extended binary perfect (16,4,2¹¹) codes is that on such codes of rank 15 over F ₂.     

Binary Extended Perfect Codes of Length 16 by the Generalized Concatenated Construction

We enumerate binary extended nonlinear perfect codes of length 16 obtained by the generalized concatenated construction (GC-construction). There are 15 different types of such codes. They are defined by pairs of MDS codes A _i : (4, 2, 64)₄. For every pair, we give the number of nonequivalent codes of this type. In total, there are 285 nonequivalent binary extended nonlinear perfect codes of length 16 obtained by the GC-construction, including the Hamming (i.e., linear) code. Thus, we obtain all binary extended perfect codes of length 16 and rank 13. Their total number is equal to 272.     

Binary Perfect Codes of Length 15 by the Generalized Concatenated Construction

We enumerate binary nonlinear perfect codes of length 15 obtained by the generalized concatenated (GC) construction. There are 15 different types of such codes. They are defined by pairs of MDS codes A _i : (4, 2, 64)₄. For every pair we give the number of nonequivalent codes of this type. In total, there are 777 nonequivalent binary perfect codes of length 15 obtained by the GC construction. This number includes the Hamming code (of rank 11), 18 Vasil'ev codes (of rank 12), and 758 codes of rank 13.     

Vasil’ev codes of length n = 2m and doubling of Steiner systems S(n, 4, 3) of a given rank

Extended binary perfect nonlinear Vasil’ev codes of length n = 2^m and Steiner systems S(n, 4, 3) of rank n-m over F ₂ are studied. The generalized concatenated construction of Vasil’ev codes induces a variant of the doubling construction for Steiner systems S(n, 4, 3) of an arbitrary rank r over F ₂. We prove that any Steiner system S(n = 2^m, 4, 3) of rank n-m can be obtained by this doubling construction and is formed by codewords of weight 4 of these Vasil’ev codes. The length 16 is studied in detail. Orders of the full automorphism groups of all 12 nonequivalent Vasil’ev codes of length 16 are found. There are exactly 15 nonisomorphic systems S(16, 4, 3) of rank 12 over F ₂, and they can be obtained from codewords of weight 4 of the extended Vasil’ev codes. Orders of the automorphism groups of all these Steiner systems are found.     

On switching equivalence of <Emphasis Type="Italic">n</Emphasis>-ary quasigroups of order 4 and perfect binary codes

We prove that arbitrary n-ary quasigroups of order 4 can be transformed into each other by successive switchings of {a, b}-components. We prove that perfect (closely packed) binary codes with distance 3 whose rank (dimension of the linear span) is greater by 1 or 2 than the rank of a linear perfect code can be taken to each other by successive switchings of i-components.     

On the Ranks and Kernels Problem for Perfect Codes

A construction is proposed which, for n large enough, allows one to build perfect binary codes of length n and rank r, with kernel of dimension k, for any admissible pair (r, k) within the limits of known bounds.     

On partitions of an <Emphasis Type="Italic">n</Emphasis>-cube into nonequivalent perfect codes

We prove that for all n = 2^k ? 1, k ≥ 5, there exists a partition of the set of all binary vectors of length n into pairwise nonequivalent perfect binary codes of length n with distance 3.     

Binary generalized (L,G) codes that are perfect in a weighted hamming metric

We propose a class of binary generalized (L,G) codes that are perfect in a weighted Hamming metric.     

Inductive Constructions of Perfect Ternary Constant-Weight Codes with Distance 3

We propose inductive constructions of perfect (n,3;n – 1)₃ codes (ternary constant-weight codes of length n and weight n – 1 with distance 3), which are modifications of constructions of perfect binary codes. The construction yields at least different perfect (n,3;n – 1)₃ codes. To perfect (n,3;n – 1)₃ codes, perfect matchings in a binary hypercube without close (at distance 1 or 2 from each other) parallel edges are equivalent.     

Construction of perfect q-ary codes by switchings of simple components

We suggest a construction of perfect q-ary codes by sequential switchings of special-type components (called simple components) of the Hamming code. We prove that such components are minimal. The construction yields a lower bound on the number of different q-ary codes; this is presently the best known bound. We show that this bound cannot be substantially improved using switchings of components of this type.     

Quotients of Gaussian graphs and their application to perfect codes

A graph-based model of perfect two-dimensional codes is presented in this work. This model facilitates the study of the metric properties of the codes. Signal spaces are modeled by means of Cayley graphs defined over the Gaussian integers and denoted as Gaussian graphs. Codewords of perfect codes will be represented by vertices of a quotient graph of the Gaussian graph in which the signal space has been defined. It will be shown that any quotient graph of a Gaussian graph is indeed a Gaussian graph. This makes it possible to apply previously known properties of Gaussian graphs to the analysis of perfect codes. To illustrate the modeling power of this graph-based tool, perfect Lee codes will be analyzed in terms of Gaussian graphs and their quotients.     

On the Structure of Symmetry Groups of Vasil’ev Codes

The structure of symmetry groups of Vasil’ev codes is studied. It is proved that the symmetry group of an arbitrary perfect binary non-full-rank Vasil’ev code of length n is always nontrivial; for codes of rank n − log(n + 1) +1, an attainable upper bound on the order of the symmetry group is obtained.__________Translated from Problemy Peredachi Informatsii, No. 2, 2005, pp. 42–49.Original Russian Text Copyright © 2005 by Avgustinovich, Solov’eva, Heden.Supported in part by the Royal Swedish Academy of Sciences.     

On the constructions of constant-composition codes from perfect nonlinear functions

A new construction of constant-composition codes based on all known perfect nonlinear functions from F _q m to itself is presented, which provides a kind of unified constructions of constant-composition codes based on all known perfect nonlinear functions from F _q m to itself. It is proved that the new constant-composition codes are optimal with respect to the Luo-Fu-Vinck-Chen bound, when m is an odd positive integer greater than 1. Finally, we point out that two constructions of constant-composition codes, proposed by Ding Cunsheng et al. in 2005, are equivalent to two special types of the new constant-composition codes. Supported in part by the National Natural Science Foundation of China (Grant Nos. 60573028, 60803156), the Open Research Fund of the National Mobile Communications Research Laboratory of Southeast University (Grant No. W200805) and in part by Singapore Ministry of Education Academic Research Fund (Grant No. T206B2204)     

New construction of perfect sequence set and low correlation zone sequence set

For a given binary ideal autocorrelation sequence, we construct a perfect sequence set by changing a few bits of the sequence. The set has a large size with respect to the period of its sequences. Based on the constructed perfect sequence set, a new class of low correlation zone sequence sets whose low correlation zone length can be chosen flexibly is obtained. Moreover, the new constructed low correlation zone sequence sets can attain Tang-Fan-Matsufuji＇s bound with suitably chosen parameters.     

On (Partial) unit memory codes based on Gabidulin codes

(Partial) unit memory ((P)UM) codes provide a powerful possibility to construct convolutional codes based on block codes in order to achieve a high decoding performance. In this contribution, a construction based on Gabidulin codes is considered. This construction requires a modified rank metric, the so-called sum rank metric. For the sum rank metric, the free rank distance, extended row rank distance, and its slope are defined analogous to the extended row distance in the Hamming metric. Upper bounds for the free rank distance and slope of (P)UM codes in the sum rank metric are derived, and an explicit construction of (P)UM codes based on Gabidulin codes is given.     

On perfect codes for an additive channel

We construct a class of perfect codes for an additive channel. The class contains classical Hamming codes.     

对偶距离为5的极大自正交码及其子码

研究了自对偶码与其删截得到的极大自正交码的等价性问题。利用删截法构造出码长n满足21≤n≤29、对偶距离为5的二元极大自正交码。再用随机搜索算法研究了所得到的二元极大自正交码的子码,构造出它们的对偶距离为3和5的子码的生成矩阵。研究了这些子码构成的码链以及它们的对偶码构成的码链。利用所得到的码链,由Steane构造法构造出距离为5的具有很好参数的量子纠错码。     

有19-(4,f)型自同构的二元自对偶码

应用二元自对偶码可看成几个自对偶码的直和理论,研究了具有19-(4,f)型自同构、码长在100以内的的二元自对偶码。这种对偶码都可看成一个码长为4的收缩码和GF(2)n上一些偶重量多项式的直和。证明了码长大于80且小于100时,不存在19-(4,f)型的二元自对偶码。根据码长较短的自对偶码分别构造出了码长为76、78和80的二元自对偶码,并给出其生成矩阵。由码的等价得到了这几类码可能的分类情况。运行Matlab程序,证明了具有19-(4,2)型和19-(4,4)型的二元自对偶码在等价情况下都有11个,19-(4,0)型的二元自对偶码在等价情况下是不存在的。     

Rank subcodes in multicomponent network coding

A new class of subcodes in rank metric is proposed; based on it, multicomponent network codes are constructed. Basic properties of subspace subcodes are considered for the family of rank codes with maximum rank distance (MRD codes). It is shown that nonuniformly restricted rank subcodes reach the Singleton bound in a number of cases. For the construction of multicomponent codes, balanced incomplete block designs and matrices in row-reduced echelon form are used. A decoding algorithm for these network codes is proposed. Examples of codes with seven and thirteen components are given.     

距离为6的二元自对偶码的子码

用随机搜索算法研究了码长n满足22≤n≤30且距离为6的二元自对偶码的子码,构造出它们的对偶距离为3、4、5和6的子码的生成矩阵。研究了这些子码构成的码链以及它们的对偶码构成的码链。利用所得到的码链,由Steane构造法构造出距离为5和6的具有很好参数的量子纠错码,改进了前人得到的几个量子纠错码的参数。