Research on the Construction of Maximum Distance Separable Codes via Arbitrary Twisted Generalized Reed-Solomon Codes

IF 2.9 3区计算机科学 Q3 COMPUTER SCIENCE, INFORMATION SYSTEMS

IEEE Transactions on Information Theory Pub Date : 2025-04-23 DOI:10.1109/TIT.2025.3563664

Chun'e Zhao;Wenping Ma;Tongjiang Yan;Yuhua Sun

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引用次数: 0

Abstract

Maximum distance separable (MDS) codes have significant combinatorial and cryptographic applications due to their certain optimality. Generalized Reed-Solomon (GRS) codes are the most prominent MDS codes. Twisted generalized Reed-Solomon (TGRS) codes may not necessarily be MDS. It is meaningful to study the conditions under which TGRS codes are MDS. In this paper, we study a general class of TGRS (A-TGRS) codes which include all the known special ones. First, we obtain another expression of the inverse of the Vandermonde matrix. Based on this, we further derive an equivalent condition under which an A-TGRS code is MDS. According to this, the A-TGRS MDS codes include nearly all the known related results in the previous literatures. More importantly, we also give three constructions to obtain many other classes of MDS TGRS codes with new parameter matrices. In addition, we present a new method to compute the inverse of the lower triangular Toplitz matrix by a linear feedback shift register, which will be very useful in many research fields.

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利用任意扭曲广义Reed-Solomon码构造最大距离可分码的研究

最大距离可分离码由于具有一定的最优性，在组合和密码学中有着重要的应用。广义Reed-Solomon （GRS）码是最突出的MDS码。扭曲广义Reed-Solomon （TGRS）码不一定是MDS。研究TGRS码是MDS的条件是有意义的。本文研究了一类一般的TGRS码（a -TGRS），它包含了所有已知的特殊码。首先，我们得到了Vandermonde矩阵逆的另一个表达式。在此基础上，进一步推导出A-TGRS码为MDS的等价条件。由此可见，A-TGRS MDS编码几乎囊括了以往文献中已知的所有相关结果。更重要的是，我们还给出了三种构造，以获得具有新参数矩阵的许多其他类型的MDS TGRS码。此外，我们还提出了一种利用线性反馈移位寄存器计算下三角形Toplitz矩阵逆的新方法，这将在许多研究领域中发挥重要作用。

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

IEEE Transactions on Information Theory 工程技术-工程：电子与电气

CiteScore

5.70

自引率

20.00%

发文量

514

审稿时长

12 months

期刊介绍： The IEEE Transactions on Information Theory is a journal that publishes theoretical and experimental papers concerned with the transmission, processing, and utilization of information. The boundaries of acceptable subject matter are intentionally not sharply delimited. Rather, it is hoped that as the focus of research activity changes, a flexible policy will permit this Transactions to follow suit. Current appropriate topics are best reflected by recent Tables of Contents; they are summarized in the titles of editorial areas that appear on the inside front cover.