The weight hierarchies of linear codes from simplicial complexes

IF 0.7 3区数学 Q2 MATHEMATICS Discrete Mathematics Pub Date : 2024-09-12 DOI:10.1016/j.disc.2024.114240

Chao Liu, Dabin Zheng , Wei Lu, Xiaoqiang Wang

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Abstract

The study of the generalized Hamming weight of linear codes is a significant research topic in coding theory as it conveys the structural information of the codes and determines their performance in various applications. However, determining the generalized Hamming weights of linear codes, especially the weight hierarchy, is generally challenging. In this paper, we investigate the generalized Hamming weights of a class of linear code $C$ over $F_{q}$ , which is constructed from defining sets. These defining sets are either special simplicial complexes or their complements in $F_{q}^{m}$ . We determine the complete weight hierarchies of these codes by analyzing the maximum or minimum intersection of certain simplicial complexes and all r-dimensional subspaces of $F_{q}^{m}$ , where $1 \leq r \leq \dim_{F_{q}} (C)$ .

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来自简单复合物的线性编码的权重层次

线性编码的广义汉明权重传递了编码的结构信息，决定了编码在各种应用中的性能，因此研究线性编码的广义汉明权重是编码理论中的一个重要研究课题。然而，确定线性编码的广义汉明权重，尤其是权重层次结构，通常具有挑战性。本文研究了一类 Fq 上线性编码 C 的广义汉明权重，该编码由定义集构建。这些定义集要么是特殊的单纯复数，要么是它们在 Fqm 中的补集。我们通过分析某些单纯复数与 Fqm 的所有 r 维子空间的最大或最小交集（其中 1≤r≤dimFq(C) ），确定这些代码的完整权重等级。

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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.

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