{"title":"借助循环多项式的循环局部可恢复液晶编码","authors":"Anuj Kumar Bhagat, Ritumoni Sarma","doi":"10.1016/j.ffa.2024.102519","DOIUrl":null,"url":null,"abstract":"<div><div>This article explores two families of cyclic codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of length <em>n</em> denoted by <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn>1</mn></mrow></msub></math></span>, which are generated by the <em>n</em>-th cyclotomic polynomial <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and the polynomial <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, respectively. We find formulae for the distance of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn>1</mn></mrow></msub></math></span> for each <span><math><mi>n</mi><mo>></mo><mn>1</mn></math></span> and conjecture formulae for the distance of their (Euclidean) duals. We prove the conjecture when <em>n</em> is a product of at most two distinct prime powers. Moreover, we show that all these codes are LCD codes, and several subfamilies are both <em>r</em>-optimal and <em>d</em>-optimal locally recoverable codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Cyclic locally recoverable LCD codes with the help of cyclotomic polynomials\",\"authors\":\"Anuj Kumar Bhagat, Ritumoni Sarma\",\"doi\":\"10.1016/j.ffa.2024.102519\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This article explores two families of cyclic codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of length <em>n</em> denoted by <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn>1</mn></mrow></msub></math></span>, which are generated by the <em>n</em>-th cyclotomic polynomial <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and the polynomial <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, respectively. We find formulae for the distance of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn>1</mn></mrow></msub></math></span> for each <span><math><mi>n</mi><mo>></mo><mn>1</mn></math></span> and conjecture formulae for the distance of their (Euclidean) duals. We prove the conjecture when <em>n</em> is a product of at most two distinct prime powers. Moreover, we show that all these codes are LCD codes, and several subfamilies are both <em>r</em>-optimal and <em>d</em>-optimal locally recoverable codes.</div></div>\",\"PeriodicalId\":50446,\"journal\":{\"name\":\"Finite Fields and Their Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-10-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Finite Fields and Their Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1071579724001588\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1071579724001588","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
本文探讨了长度为 n 的 Fq 上的两个循环码族,分别用 Cn 和 Cn,1 表示,它们分别由 n 次循环多项式 Qn(x) 和多项式 Qn(x)Q1(x) 生成。我们找到了每个 n>1 的 Cn 和 Cn,1 的距离公式,并猜想了它们(欧几里得)对偶的距离公式。当 n 是最多两个不同质幂的乘积时,我们证明了猜想。此外,我们还证明了所有这些编码都是 LCD 编码,而且有几个子系列既是 r-最优编码,也是 d-最优局部可恢复编码。
Cyclic locally recoverable LCD codes with the help of cyclotomic polynomials
This article explores two families of cyclic codes over of length n denoted by and , which are generated by the n-th cyclotomic polynomial and the polynomial , respectively. We find formulae for the distance of and for each and conjecture formulae for the distance of their (Euclidean) duals. We prove the conjecture when n is a product of at most two distinct prime powers. Moreover, we show that all these codes are LCD codes, and several subfamilies are both r-optimal and d-optimal locally recoverable codes.
期刊介绍:
Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering.
For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods.
The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.