Several families of negacyclic BCH codes and their duals

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

Designs, Codes and Cryptography Pub Date : 2024-12-27 DOI:10.1007/s10623-024-01551-2

Zhonghua Sun, Xinyue Liu

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引用次数: 0

Abstract

Negacyclic BCH codes are a special subclasses of negacyclic codes, and have the best parameters known in many cases. A family of good negacyclic BCH codes are the q-ary narrow-sense negacyclic BCH codes of length \(n=(q^m-1)/2\), where q is an odd prime power. Little is known about the true minimum distance of this family of negacyclic BCH codes and the dimension of this family of negacyclic BCH codes with large designed distance. The main objective of this paper is to study three subfamilies of this family of negacyclic BCH codes. The dimension and true minimum distance of a subfamily of the q-ary narrow-sense negacyclic BCH codes of length n are determined. The dimension and good lower bounds on the minimum distance of two subfamilies of the q-ary narrow-sense negacyclic BCH codes of length n are presented. The minimum distances of the duals of the q-ary narrow-sense negacyclic BCH codes of length n are also investigated. As will be seen, the three subfamilies of negacyclic BCH codes are sometimes distance-optimal and sometimes have the same parameters as the best linear codes known.

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几个负环BCH码族及其对偶

负环BCH码是负环码的一个特殊子类，在许多情况下具有已知的最佳参数。一类好的负环BCH码是长度为\(n=(q^m-1)/2\)的q元狭义负环BCH码，其中q是奇素数幂。对于这类负环BCH码的真实最小距离和设计距离较大的这类负环BCH码的维数，我们知之甚少。本文的主要目的是研究这个负循环BCH码族的三个亚族。确定了长度为n的q元狭义负环BCH码的一个亚族的维数和真最小距离。给出了长度为n的q元狭义负环BCH码的两个亚族的最小距离的维数和良好的下界。研究了长度为n的q元窄意义负环BCH码对偶的最小距离。正如将看到的，负循环BCH码的三个亚族有时是距离最优的，有时与已知的最佳线性码具有相同的参数。

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.