Griesmer type bounds for additive codes over finite fields, integral and fractional MDS codes

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Designs, Codes and Cryptography Pub Date : 2024-11-03 DOI:10.1007/s10623-024-01519-2

Simeon Ball, Michel Lavrauw, Tabriz Popatia

引用次数: 0

Abstract

In this article we prove Griesmer type bounds for additive codes over finite fields. These new bounds give upper bounds on the length of maximum distance separable (MDS) codes, codes which attain the Singleton bound. We will also consider codes to be MDS if they attain the fractional Singleton bound, due to Huffman. We prove that this bound in the fractional case can be obtained by codes whose length surpasses the length of the longest known codes in the integral case. For small parameters, we provide exhaustive computational results for additive MDS codes, by classifying the corresponding (fractional) subspace-arcs. This includes a complete classification of fractional additive MDS codes of size 243 over the field of order 9.

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有限域上加法码、积分和分数 MDS 码的格里斯梅尔类型界限

在这篇文章中，我们证明了有限域上加法码的格里斯梅尔类型界限。这些新边界给出了最大距离可分码（MDS）长度的上限，即达到 Singleton 边界的码。如果代码达到了哈夫曼提出的分数单子约束，我们也会认为它们是 MDS 代码。我们证明，在分数情况下，长度超过积分情况下已知最长编码长度的编码可以获得这一约束。对于小参数，我们通过对相应的（分数）子空间弧进行分类，为加法 MDS 编码提供了详尽的计算结果。这包括对 9 阶域上大小为 243 的分数加法 MDS 码的完整分类。

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

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.