{"title":"一类长度为 2(q2m-1)q+1 的 BCH 码及其对偶码","authors":"Huan Zhu, Jin Li, Shixin Zhu","doi":"10.1016/j.disc.2024.114152","DOIUrl":null,"url":null,"abstract":"<div><p>BCH codes are a special subclass of cyclic codes. In many cases, BCH codes are best cyclic codes and they have wide applications in communication and storage systems. In this paper, we investigate the parameters of a class of narrow-sense BCH codes over <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> of length <span><math><mfrac><mrow><mn>2</mn><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span> with small and large dimensions. We study the <em>q</em>-cyclotomic cosets modulo <span><math><mfrac><mrow><mn>2</mn><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span>, determine the dimensions of these BCH codes and give the lower bounds on their minimum distances. Furthermore, we present the lower bounds on the minimum distances of their dual codes.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A class of BCH codes of length 2(q2m−1)q+1 and their duals\",\"authors\":\"Huan Zhu, Jin Li, Shixin Zhu\",\"doi\":\"10.1016/j.disc.2024.114152\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>BCH codes are a special subclass of cyclic codes. In many cases, BCH codes are best cyclic codes and they have wide applications in communication and storage systems. In this paper, we investigate the parameters of a class of narrow-sense BCH codes over <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> of length <span><math><mfrac><mrow><mn>2</mn><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span> with small and large dimensions. We study the <em>q</em>-cyclotomic cosets modulo <span><math><mfrac><mrow><mn>2</mn><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span>, determine the dimensions of these BCH codes and give the lower bounds on their minimum distances. Furthermore, we present the lower bounds on the minimum distances of their dual codes.</p></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-07-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X24002838\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24002838","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
A class of BCH codes of length 2(q2m−1)q+1 and their duals
BCH codes are a special subclass of cyclic codes. In many cases, BCH codes are best cyclic codes and they have wide applications in communication and storage systems. In this paper, we investigate the parameters of a class of narrow-sense BCH codes over of length with small and large dimensions. We study the q-cyclotomic cosets modulo , determine the dimensions of these BCH codes and give the lower bounds on their minimum distances. Furthermore, we present the lower bounds on the minimum distances of their dual codes.
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.