The extended codes of some linear codes

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2024-03-12 DOI:10.1016/j.ffa.2024.102401

Zhonghua Sun , Cunsheng Ding , Tingfang Chen

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引用次数: 0

Abstract

The classical way of extending an $[n, k, d]$ linear code $C$ is to add an overall parity-check coordinate to each codeword of the linear code $C$ . This extended code, denoted by $\overline{C} (- 1)$ and called the standardly extended code of $C$ , is a linear code with parameters $[n + 1, k, \bar{d}]$ , where $\bar{d} = d$ or $\bar{d} = d + 1$ . This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper.

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某些线性编码的扩展编码

扩展[n,k,d]线性码 C 的经典方法是在线性码 C 的每个码字上添加一个总体奇偶校验坐标。这种扩展码用 C‾(-1) 表示，称为 C 的标准扩展码，是参数为 [n+1,k,d¯] 的线性码，其中 d¯=d 或 d¯=d+1 。这是文献中线性编码的两种扩展技术之一。一些二进制线性编码族的标准扩展编码已在一定程度上得到研究。然而，人们对非二进制码的标准扩展码还知之甚少。例如，非二进制汉明码的标准扩展码的最小距离问题 70 多年来一直悬而未决。本文的第一个目的是介绍线性编码的非标准扩展编码，并为这类扩展线性编码建立一些一般理论。第二个目的是研究若干线性码族的这类扩展码，包括循环码和非二进制汉明码。利用这种扩展技术得到了四个距离最优或维度最优的线性码族。本文解决了多个线性码族的某些扩展码的参数问题。

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

Finite Fields and Their Applications 数学-数学

CiteScore

2.00

自引率

20.00%

发文量

133

审稿时长

6-12 weeks

期刊介绍： Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.