{"title":"有三个零的几类最优循环码","authors":"Tingting Wu, Li Liu, Lanqiang Li","doi":"10.1007/s00200-023-00636-0","DOIUrl":null,"url":null,"abstract":"<p>As a class of linear codes, cyclic codes are widely used in communication systems, consumer electronics and data storage systems due to their favorable properties. In this paper, we construct two classes of optimal <i>p</i>-ary cyclic codes with parameters <span>\\([p^m-1, p^m-\\frac{3m}{2}-2, 4]\\)</span> by analyzing the solutions of certain polynomials over finite fields. Furthermore, we propose an efficient method to determine the optimality of the 7-ary cyclic code <span>\\(\\mathcal {C}_{(0,1,e)}\\)</span> and present three classes of optimal codes with parameters <span>\\([7^m-1,7^m-2m-2,4]\\)</span>. Additionally, we provide the weight distribution of one class of their duals.</p>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"58 11 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Several classes of optimal cyclic codes with three zeros\",\"authors\":\"Tingting Wu, Li Liu, Lanqiang Li\",\"doi\":\"10.1007/s00200-023-00636-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>As a class of linear codes, cyclic codes are widely used in communication systems, consumer electronics and data storage systems due to their favorable properties. In this paper, we construct two classes of optimal <i>p</i>-ary cyclic codes with parameters <span>\\\\([p^m-1, p^m-\\\\frac{3m}{2}-2, 4]\\\\)</span> by analyzing the solutions of certain polynomials over finite fields. Furthermore, we propose an efficient method to determine the optimality of the 7-ary cyclic code <span>\\\\(\\\\mathcal {C}_{(0,1,e)}\\\\)</span> and present three classes of optimal codes with parameters <span>\\\\([7^m-1,7^m-2m-2,4]\\\\)</span>. Additionally, we provide the weight distribution of one class of their duals.</p>\",\"PeriodicalId\":50742,\"journal\":{\"name\":\"Applicable Algebra in Engineering Communication and Computing\",\"volume\":\"58 11 1\",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-12-07\",\"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/s00200-023-00636-0\",\"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/s00200-023-00636-0","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}
Several classes of optimal cyclic codes with three zeros
As a class of linear codes, cyclic codes are widely used in communication systems, consumer electronics and data storage systems due to their favorable properties. In this paper, we construct two classes of optimal p-ary cyclic codes with parameters \([p^m-1, p^m-\frac{3m}{2}-2, 4]\) by analyzing the solutions of certain polynomials over finite fields. Furthermore, we propose an efficient method to determine the optimality of the 7-ary cyclic code \(\mathcal {C}_{(0,1,e)}\) and present three classes of optimal codes with parameters \([7^m-1,7^m-2m-2,4]\). Additionally, we provide the weight distribution of one class of their duals.
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.
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