A New Upper Bound for Linear Codes and Vanishing Partial Weight Distributions

IF 2.2 3区 计算机科学 Q3 COMPUTER SCIENCE, INFORMATION SYSTEMS IEEE Transactions on Information Theory Pub Date : 2024-08-26 DOI:10.1109/TIT.2024.3449899
Hao Chen;Conghui Xie
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引用次数: 0

Abstract

In this paper, we give a new upper bound on sizes of linear codes related to weight distributions of codes as follows. Let C be a linear $[n,k,d]_{q}$ code, such that, between d and $d\left ({{1+\frac {1}{q-1}}}\right)-1$ , the largest weight of codewords in C is the weight $d\left ({{1+\frac {1}{q-1}}}\right)-1-v$ , then $k \leq n-d\left ({{1+\frac {1}{q-1}}}\right)+2+v$ . Some infinite families of linear codes with arbitrary minimum distances attaining this bound are constructed. This bound is stronger than the Singleton bound for linear codes. Hence we prove that there is no codeword of weights in the range $\left [{{\frac {qd}{q-1}-v,\frac {qd}{q-1}-1}}\right]$ for a linear $[n,k,d]_{q}$ code, if $v=\frac {qd}{q-1}+k-n-2 \geq 2$ . This is the first such kind of result, which concludes vanishing partial weight distributions from four parameters $n,k,d$ and q. Then we give vanishing partial weight distribution results for many best known linear codes, some almost MDS codes, general small Griesmer defect codes, some BCH codes, and some cyclic codes. Upper bounds on the number of nonzero weights of binary Griesmer codes and some small Singleton defect codes are also given.
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线性编码和消失部分权重分布的新上限
本文给出了与码权分布有关的线性码的大小的一个新的上界。设C为线性$[n,k,d]_{q}$码,使得在d和$d\left ({{1+\frac {1}{q-1}}}\right)-1$之间,C中码字的最大权值为$d\left ({{1+\frac {1}{q-1}}}\right)-1-v$,然后为$k \leq n-d\left ({{1+\frac {1}{q-1}}}\right)+2+v$。构造了具有任意最小距离的无限族线性码。对于线性代码,此界比单例界强。因此,我们证明了对于一个线性的$[n,k,d]_{q}$码,如果$v=\frac {qd}{q-1}+k-n-2 \geq 2$,在$\left [{{\frac {qd}{q-1}-v,\frac {qd}{q-1}-1}}\right]$范围内不存在权码字。这是第一个这样的结果,得出了四个参数$n,k,d$和q的消失偏权分布。然后我们给出了许多最著名的线性码、一些几乎MDS码、一般小Griesmer缺陷码、一些BCH码和一些循环码的消失偏权分布结果。给出了二元Griesmer码和一些较小的单缺陷码的非零权数的上界。
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
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory 工程技术-工程:电子与电气
CiteScore
5.70
自引率
20.00%
发文量
514
审稿时长
12 months
期刊介绍: The IEEE Transactions on Information Theory is a journal that publishes theoretical and experimental papers concerned with the transmission, processing, and utilization of information. The boundaries of acceptable subject matter are intentionally not sharply delimited. Rather, it is hoped that as the focus of research activity changes, a flexible policy will permit this Transactions to follow suit. Current appropriate topics are best reflected by recent Tables of Contents; they are summarized in the titles of editorial areas that appear on the inside front cover.
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