A class of BCH codes with length $$\frac{q^{2m}-1}{q+1}$$

IF 0.6 4区工程技术 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-12-27 DOI:10.1007/s00200-023-00637-z

引用次数: 0

Abstract

As an important class of cyclic codes, BCH codes are widely employed in satellite communications, DVDs, CD, DAT etc. In this paper, we determine the dimension of BCH codes of length $\frac{q^{2m}-1}{q+1}$ over the finite fields ${\mathbb {F}}_q$ . We settle a conjecture about the largest q-cyclotomic coset leader modulo n which was proposed by Wu et al. We also get the second largest q-cyclotomic coset leader modulo n if m is odd. Moreover, we investigate the parameters of ${\mathcal {C}}_{(n,q,\delta _i)}$ ( ${\mathcal {C}}_{(n,q,\delta _i)}^\perp$ ) for $i=1,2$ and ${\mathcal {C}}_{(n,q,\delta )}$ ( ${\mathcal {C}}_{(n,q,\delta )}^\perp$ ) for $2\le \delta \le q-1$ . What’s more, we obtain many (almost) optimal codes.

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一类长度为 $$frac{q^{2m}-1}{q+1}$ 的 BCH 编码

摘要 BCH码作为一类重要的循环码，被广泛应用于卫星通信、DVD、CD、DAT等领域。本文确定了有限域 ${\mathbb {F}}_q$ 上长度为 $\frac{q^{2m}-1}{q+1}$ 的 BCH 码的维数。我们解决了 Wu 等人提出的关于最大 q-cyclotomic coset leader modulo n 的猜想，并得到了当 m 为奇数时第二大 q-cyclotomic coset leader modulo n。此外，我们研究了 ${\mathcal {C}}_{(n,q,\delta _i)}$ ( ${\mathcal {C}}_{(n,q,\delta _i)}^\perp$ ) 对于 $i=1、2) and\({\mathcal {C}}_{(n,q,\delta )}$ ( ${\mathcal {C}}_{(n,q,\delta )}^\perp$ ) for$2\le \delta \le q-1$ .此外，我们还得到了许多（几乎）最优编码。

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

Applicable Algebra in Engineering Communication and Computing 工程技术-计算机：跨学科应用

CiteScore

2.90

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Algebra is a common language for many scientific domains. In developing this language mathematicians prove theorems and design methods which demonstrate the applicability of algebra. Using this language scientists in many fields find algebra indispensable to create methods, techniques and tools to solve their specific problems. Applicable Algebra in Engineering, Communication and Computing will publish mathematically rigorous, original research papers reporting on algebraic methods and techniques relevant to all domains concerned with computers, intelligent systems and communications. Its scope includes, but is not limited to, vision, robotics, system design, fault tolerance and dependability of systems, VLSI technology, signal processing, signal theory, coding, error control techniques, cryptography, protocol specification, networks, software engineering, arithmetics, algorithms, complexity, computer algebra, programming languages, logic and functional programming, algebraic specification, term rewriting systems, theorem proving, graphics, modeling, knowledge engineering, expert systems, and artificial intelligence methodology. Purely theoretical papers will not primarily be sought, but papers dealing with problems in such domains as commutative or non-commutative algebra, group theory, field theory, or real algebraic geometry, which are of interest for applications in the above mentioned fields are relevant for this journal. On the practical side, technology and know-how transfer papers from engineering which either stimulate or illustrate research in applicable algebra are within the scope of the journal.