On two conjectures of the duals of AMDS BCH codes

IF 1.3 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2026-03-01 Epub Date: 2025-12-01 DOI:10.1016/j.ffa.2025.102765

Haode Yan

引用次数: 0

Abstract

In this paper, we study the duals of BCH codes

C_{(q, q + 1, 3, \frac{p - 1}{2})}

and

C_{(3^{s}, 3^{s} + 1, 3, 4)}

. By analyzing the number of solutions to certain equations over the multiplicative subgroup

U_{q + 1} = {x \in F_{q^{2}} : x^{q + 1} = 1}

F_{q^{2}}^{⁎}

, we determine the possible weights of codewords in

C_{(q, q + 1, 3, \frac{p - 1}{2})}^{⊥}

and

C_{(3^{s}, 3^{s} + 1, 3, 4)}^{⊥}

, respectively. The weight distributions of these two dual codes are derived by applying the Pless power moments. Our results provide affirmative solutions to recent conjectures.

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关于AMDS BCH码对偶的两个猜想

本文研究了BCH码C（q,q+1,3,p−12）和C（3s,3s+1,3,4）的对偶。通过分析Fq2的乘法子群Uq+1={x∈Fq2:xq+1=1}上某些方程的解的个数，我们分别确定了C(q,q+1,3,p−12)⊥和C(3s,3s+1,3,4)⊥中码字的可能权重。应用无功矩导出了这两种双码的权值分布。我们的结果为最近的猜想提供了肯定的答案。

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.