Constructing MRD codes by switching

IF 0.5 4区数学 Q3 MATHEMATICS Journal of Combinatorial Designs Pub Date : 2024-02-08 DOI:10.1002/jcd.21931

Minjia Shi, Denis S. Krotov, Ferruh Özbudak

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引用次数: 0

Abstract

Maximum rank-distance (MRD) codes are (not necessarily linear) maximum codes in the rank-distance metric space on $m$ -by- $n$ matrices over a finite field $F_{q}$ . They are diameter perfect and have the cardinality $q^{m (n - d + 1)}$ if $m \geq n$ . We define switching in MRD codes as the replacement of special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting switching, such as punctured twisted Gabidulin codes and direct-product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in $m$ if the other parameters ( $n, q$ , the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes.

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通过切换构建 MRD 代码

最大秩距（MRD）编码是有限域 Fq${{mathbb{F}}}_{q}$ 上 m$m$-by-n$n$ 矩阵的秩距度量空间中的（不一定是线性的）最大编码。如果 m≥n$m\ge n$，它们的直径是完美的，并且具有 qm(n-d+1)${q}^{m(n-d+1)}$ 的心数。我们将 MRD 编码中的转换定义为用参数相同的其他子编码替换特殊的 MRD 子编码。我们考虑了允许交换的 MRD 码的构造，如点状扭曲加比杜林码和直积码。利用切换，我们构造了一大类 MRD 码，如果其他参数（n,q,q,n,\,q$，码距）固定不变，这些码的心数在 m$m$ 中以双指数形式增长。此外，我们还构造了不同仿射等级的 MRD 码和非周期性 MRD 码。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Combinatorial Designs 数学-数学

CiteScore

1.60

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including: block designs, t-designs, pairwise balanced designs and group divisible designs Latin squares, quasigroups, and related algebras computational methods in design theory construction methods applications in computer science, experimental design theory, and coding theory graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics finite geometry and its relation with design theory. algebraic aspects of design theory. Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.