{"title":"关于循环码汉明距离的一些结果","authors":"Guantao Pan, Lanqiang Li, Ziwen Cao, Fuyin Tian","doi":"10.1007/s00200-024-00660-8","DOIUrl":null,"url":null,"abstract":"<p>Cyclic codes over finite fields have been studied for decades due to their wide applicability in communication systems, consumer electronics, and data storage systems. Let <i>p</i> be an odd prime and let <i>s</i> and <i>m</i> be positive integers. In this paper, we first determine the Hamming distances of all cyclic codes of length 8 over <span>\\(F_q\\)</span>. Building upon this, we explicitly obtain the Hamming distances of all repeated-root cyclic codes of length <span>\\(8p^s\\)</span> over <span>\\(F_q\\)</span>. As an application, we give all maximum distance separable cyclic codes of length <span>\\(8p^s\\)</span>.</p>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"22 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Some results on the Hamming distances of cyclic codes\",\"authors\":\"Guantao Pan, Lanqiang Li, Ziwen Cao, Fuyin Tian\",\"doi\":\"10.1007/s00200-024-00660-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Cyclic codes over finite fields have been studied for decades due to their wide applicability in communication systems, consumer electronics, and data storage systems. Let <i>p</i> be an odd prime and let <i>s</i> and <i>m</i> be positive integers. In this paper, we first determine the Hamming distances of all cyclic codes of length 8 over <span>\\\\(F_q\\\\)</span>. Building upon this, we explicitly obtain the Hamming distances of all repeated-root cyclic codes of length <span>\\\\(8p^s\\\\)</span> over <span>\\\\(F_q\\\\)</span>. As an application, we give all maximum distance separable cyclic codes of length <span>\\\\(8p^s\\\\)</span>.</p>\",\"PeriodicalId\":50742,\"journal\":{\"name\":\"Applicable Algebra in Engineering Communication and Computing\",\"volume\":\"22 1\",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-05-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applicable Algebra in Engineering Communication and Computing\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://doi.org/10.1007/s00200-024-00660-8\",\"RegionNum\":4,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applicable Algebra in Engineering Communication and Computing","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1007/s00200-024-00660-8","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
摘要
由于其在通信系统、消费电子产品和数据存储系统中的广泛应用，有限域上的循环码已被研究了几十年。假设 p 是奇素数，s 和 m 是正整数。在本文中，我们首先确定了 \(F_q\) 上所有长度为 8 的循环码的汉明距离。在此基础上，我们明确地得到了所有长度为 \(8p^s\) over\(F_q\) 的重复根循环码的汉明距离。作为应用，我们给出了所有长度为 （8p^s\ ）的最大距离可分离循环码。
Some results on the Hamming distances of cyclic codes
Cyclic codes over finite fields have been studied for decades due to their wide applicability in communication systems, consumer electronics, and data storage systems. Let p be an odd prime and let s and m be positive integers. In this paper, we first determine the Hamming distances of all cyclic codes of length 8 over \(F_q\). Building upon this, we explicitly obtain the Hamming distances of all repeated-root cyclic codes of length \(8p^s\) over \(F_q\). As an application, we give all maximum distance separable cyclic codes of length \(8p^s\).
期刊介绍：
Algebra is a common language for many scientific domains. In developing this language mathematicians prove theorems and design methods which demonstrate the applicability of algebra. Using this language scientists in many fields find algebra indispensable to create methods, techniques and tools to solve their specific problems.
Applicable Algebra in Engineering, Communication and Computing will publish mathematically rigorous, original research papers reporting on algebraic methods and techniques relevant to all domains concerned with computers, intelligent systems and communications. Its scope includes, but is not limited to, vision, robotics, system design, fault tolerance and dependability of systems, VLSI technology, signal processing, signal theory, coding, error control techniques, cryptography, protocol specification, networks, software engineering, arithmetics, algorithms, complexity, computer algebra, programming languages, logic and functional programming, algebraic specification, term rewriting systems, theorem proving, graphics, modeling, knowledge engineering, expert systems, and artificial intelligence methodology.
Purely theoretical papers will not primarily be sought, but papers dealing with problems in such domains as commutative or non-commutative algebra, group theory, field theory, or real algebraic geometry, which are of interest for applications in the above mentioned fields are relevant for this journal.
On the practical side, technology and know-how transfer papers from engineering which either stimulate or illustrate research in applicable algebra are within the scope of the journal.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
