MacWilliams' Extension Theorem for rank-metric codes

IF 0.6 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation Pub Date : 2023-09-01 DOI:10.1016/j.jsc.2023.102263

Elisa Gorla, Flavio Salizzoni

引用次数: 0

Abstract

The MacWilliams' Extension Theorem is a classical result by Florence Jessie MacWilliams. It shows that every linear isometry between linear block-codes endowed with the Hamming distance can be extended to a linear isometry of the ambient space. Such an extension fails to exist in general for rank-metric codes, that is, one can easily find examples of linear isometries between rank-metric codes which cannot be extended to linear isometries of the ambient space. In this paper, we explore to what extent a MacWilliams' Extension Theorem may hold for rank-metric codes. We provide an extensive list of examples of obstructions to the existence of an extension, as well as a positive result.

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等级-度量码的MacWilliams可拓定理

麦克威廉姆斯可拓定理是弗洛伦斯·杰西·麦克威廉姆斯的经典结果。结果表明，赋予Hamming距离的线性分组码之间的每一个线性等距都可以推广到环境空间的线性等距。这种扩展通常不存在于秩度量码，也就是说，人们可以很容易地找到秩度量码之间的线性等距的例子，这些例子不能扩展到环境空间的线性等距。在本文中，我们探讨了MacWilliams扩展定理在多大程度上适用于秩度量码。我们提供了大量阻碍延期存在的例子，以及积极的结果。

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 Symbolic Computation 工程技术-计算机：理论方法

CiteScore

2.10

自引率

14.30%

发文量

审稿时长

142 days

期刊介绍： An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects. It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.