Some bounds on the cardinality of the b-symbol weight spectrum of codes

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2025-07-16 DOI:10.1016/j.disc.2025.114678

Hongwei Zhu , Shitao Li , Minjia Shi , Shu-Tao Xia , Patrick Solé

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引用次数: 0

Abstract

The size of the Hamming distance spectrum of a code has received great attention in recent research. The main objective of this paper is to extend these significant theories to the b-symbol distance spectrum. We examine this question for various types of codes, including unrestricted codes, additive codes, linear codes, and cyclic codes, successively. For the first three cases, we determine the maximum size of the b-symbol distance spectra of these codes smoothly. For the case of cyclic codes, we introduce three approaches to characterize the upper bound for the cardinality of the b-symbol weight spectrum of cyclic codes, namely the period distribution approach, the primitive idempotent approach, and the b-symbol weight formula approach. As two by-products of this paper, the maximum number of symplectic weights of linear codes is determined, and a basic inequality among the parameters

{[n, k, d_{H} (C)]}_{q}

of cyclic codes is provided.

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码的b符号权谱的基数的一些界

码汉明距离谱的大小是近年来研究的热点问题。本文的主要目的是将这些重要的理论推广到b符号距离谱。我们对各种类型的码，包括无限制码、加性码、线性码和循环码，依次研究了这个问题。对于前三种情况，我们平滑地确定了这些码的b符号距离谱的最大尺寸。对于循环码，我们引入了三种方法来表征循环码的b符号权谱基数的上界，即周期分布法、原始幂等法和b符号权公式法。作为本文的两个副产物，确定了线性码的最大辛权数，并给出了循环码参数[n，k,dH(C)]q之间的一个基本不等式。

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

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.