Cyclic locally recoverable LCD codes with the help of cyclotomic polynomials

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2024-10-15 DOI:10.1016/j.ffa.2024.102519

Anuj Kumar Bhagat, Ritumoni Sarma

引用次数: 0

Abstract

This article explores two families of cyclic codes over

F_{q}

of length n denoted by

C_{n}

and

C_{n, 1}

, which are generated by the n-th cyclotomic polynomial

Q_{n} (x)

and the polynomial

Q_{n} (x) Q_{1} (x)

, respectively. We find formulae for the distance of

C_{n}

and

C_{n, 1}

for each

n > 1

and conjecture formulae for the distance of their (Euclidean) duals. We prove the conjecture when n is a product of at most two distinct prime powers. Moreover, we show that all these codes are LCD codes, and several subfamilies are both r-optimal and d-optimal locally recoverable codes.

查看原文本刊更多论文

借助循环多项式的循环局部可恢复液晶编码

本文探讨了长度为 n 的 Fq 上的两个循环码族，分别用 Cn 和 Cn,1 表示，它们分别由 n 次循环多项式 Qn(x) 和多项式 Qn(x)Q1(x) 生成。我们找到了每个 n>1 的 Cn 和 Cn,1 的距离公式，并猜想了它们（欧几里得）对偶的距离公式。当 n 是最多两个不同质幂的乘积时，我们证明了猜想。此外，我们还证明了所有这些编码都是 LCD 编码，而且有几个子系列既是 r-最优编码，也是 d-最优局部可恢复编码。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

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.