关于纠错码的数目

IF 0.9 4区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Combinatorics, Probability & Computing Pub Date : 2023-06-09 DOI:10.1017/s0963548323000111

Dingding Dong, Nitya Mani, Yufei Zhao

{"title":"关于纠错码的数目","authors":"Dingding Dong, Nitya Mani, Yufei Zhao","doi":"10.1017/s0963548323000111","DOIUrl":null,"url":null,"abstract":"Abstract We show that for a fixed $q$ , the number of $q$ -ary $t$ -error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t \\leq (1 - q^{-1})n - 2\\sqrt{n \\log n}$ , where $H_q(n, t) = q^n/ V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for $t = o(n^{1/3} (\\log n)^{-2/3})$ .","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the number of error correcting codes\",\"authors\":\"Dingding Dong, Nitya Mani, Yufei Zhao\",\"doi\":\"10.1017/s0963548323000111\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We show that for a fixed $q$ , the number of $q$ -ary $t$ -error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t \\\\leq (1 - q^{-1})n - 2\\\\sqrt{n \\\\log n}$ , where $H_q(n, t) = q^n/ V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for $t = o(n^{1/3} (\\\\log n)^{-2/3})$ .\",\"PeriodicalId\":10513,\"journal\":{\"name\":\"Combinatorics, Probability & Computing\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-06-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Combinatorics, Probability & Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1017/s0963548323000111\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorics, Probability & Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/s0963548323000111","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}

引用次数: 0

摘要

摘要对于一个固定的$q$，对于所有$t \leq (1 - q^{-1})n - 2\sqrt{n \log n}$，长度为$n$的$q$ -任意$t$ -纠错码的数量最多为$2^{(1 + o(1)) H_q(n,t)}$，其中$H_q(n, t) = q^n/ V_q(n,t)$为Hamming界，$V_q(n,t)$为半径$t$的Hamming球的基数。这证明了巴洛格、特雷格伦和瓦格纳的一个猜想，他们在$t = o(n^{1/3} (\log n)^{-2/3})$上给出了结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

On the number of error correcting codes

Abstract We show that for a fixed $q$ , the number of $q$ -ary $t$ -error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t \leq (1 - q^{-1})n - 2\sqrt{n \log n}$ , where $H_q(n, t) = q^n/ V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for $t = o(n^{1/3} (\log n)^{-2/3})$ .

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Combinatorics, Probability & Computing 数学-计算机：理论方法

CiteScore

2.40

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： Published bimonthly, Combinatorics, Probability & Computing is devoted to the three areas of combinatorics, probability theory and theoretical computer science. Topics covered include classical and algebraic graph theory, extremal set theory, matroid theory, probabilistic methods and random combinatorial structures; combinatorial probability and limit theorems for random combinatorial structures; the theory of algorithms (including complexity theory), randomised algorithms, probabilistic analysis of algorithms, computational learning theory and optimisation.