{"title":"A New Upper Bound for Linear Codes and Vanishing Partial Weight Distributions","authors":"Hao Chen;Conghui Xie","doi":"10.1109/TIT.2024.3449899","DOIUrl":null,"url":null,"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<inline-formula> <tex-math>$[n,k,d]_{q}$ </tex-math></inline-formula>\n code, such that, between d and \n<inline-formula> <tex-math>$d\\left ({{1+\\frac {1}{q-1}}}\\right)-1$ </tex-math></inline-formula>\n, the largest weight of codewords in C is the weight \n<inline-formula> <tex-math>$d\\left ({{1+\\frac {1}{q-1}}}\\right)-1-v$ </tex-math></inline-formula>\n, then \n<inline-formula> <tex-math>$k \\leq n-d\\left ({{1+\\frac {1}{q-1}}}\\right)+2+v$ </tex-math></inline-formula>\n. 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 \n<inline-formula> <tex-math>$\\left [{{\\frac {qd}{q-1}-v,\\frac {qd}{q-1}-1}}\\right]$ </tex-math></inline-formula>\n for a linear \n<inline-formula> <tex-math>$[n,k,d]_{q}$ </tex-math></inline-formula>\n code, if \n<inline-formula> <tex-math>$v=\\frac {qd}{q-1}+k-n-2 \\geq 2$ </tex-math></inline-formula>\n. This is the first such kind of result, which concludes vanishing partial weight distributions from four parameters \n<inline-formula> <tex-math>$n,k,d$ </tex-math></inline-formula>\n 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.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 12","pages":"8713-8722"},"PeriodicalIF":2.2000,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"IEEE Transactions on Information Theory","FirstCategoryId":"94","ListUrlMain":"https://ieeexplore.ieee.org/document/10648774/","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, INFORMATION SYSTEMS","Score":null,"Total":0}
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.
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.