{"title":"几个负环BCH码族及其对偶","authors":"Zhonghua Sun, Xinyue Liu","doi":"10.1007/s10623-024-01551-2","DOIUrl":null,"url":null,"abstract":"<p>Negacyclic BCH codes are a special subclasses of negacyclic codes, and have the best parameters known in many cases. A family of good negacyclic BCH codes are the <i>q</i>-ary narrow-sense negacyclic BCH codes of length <span>\\(n=(q^m-1)/2\\)</span>, where <i>q</i> is an odd prime power. Little is known about the true minimum distance of this family of negacyclic BCH codes and the dimension of this family of negacyclic BCH codes with large designed distance. The main objective of this paper is to study three subfamilies of this family of negacyclic BCH codes. The dimension and true minimum distance of a subfamily of the <i>q</i>-ary narrow-sense negacyclic BCH codes of length <i>n</i> are determined. The dimension and good lower bounds on the minimum distance of two subfamilies of the <i>q</i>-ary narrow-sense negacyclic BCH codes of length <i>n</i> are presented. The minimum distances of the duals of the <i>q</i>-ary narrow-sense negacyclic BCH codes of length <i>n</i> are also investigated. As will be seen, the three subfamilies of negacyclic BCH codes are sometimes distance-optimal and sometimes have the same parameters as the best linear codes known.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"25 1","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Several families of negacyclic BCH codes and their duals\",\"authors\":\"Zhonghua Sun, Xinyue Liu\",\"doi\":\"10.1007/s10623-024-01551-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Negacyclic BCH codes are a special subclasses of negacyclic codes, and have the best parameters known in many cases. A family of good negacyclic BCH codes are the <i>q</i>-ary narrow-sense negacyclic BCH codes of length <span>\\\\(n=(q^m-1)/2\\\\)</span>, where <i>q</i> is an odd prime power. Little is known about the true minimum distance of this family of negacyclic BCH codes and the dimension of this family of negacyclic BCH codes with large designed distance. The main objective of this paper is to study three subfamilies of this family of negacyclic BCH codes. The dimension and true minimum distance of a subfamily of the <i>q</i>-ary narrow-sense negacyclic BCH codes of length <i>n</i> are determined. The dimension and good lower bounds on the minimum distance of two subfamilies of the <i>q</i>-ary narrow-sense negacyclic BCH codes of length <i>n</i> are presented. The minimum distances of the duals of the <i>q</i>-ary narrow-sense negacyclic BCH codes of length <i>n</i> are also investigated. As will be seen, the three subfamilies of negacyclic BCH codes are sometimes distance-optimal and sometimes have the same parameters as the best linear codes known.</p>\",\"PeriodicalId\":11130,\"journal\":{\"name\":\"Designs, Codes and Cryptography\",\"volume\":\"25 1\",\"pages\":\"\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-12-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Designs, Codes and Cryptography\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10623-024-01551-2\",\"RegionNum\":2,\"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":"Designs, Codes and Cryptography","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10623-024-01551-2","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Several families of negacyclic BCH codes and their duals
Negacyclic BCH codes are a special subclasses of negacyclic codes, and have the best parameters known in many cases. A family of good negacyclic BCH codes are the q-ary narrow-sense negacyclic BCH codes of length \(n=(q^m-1)/2\), where q is an odd prime power. Little is known about the true minimum distance of this family of negacyclic BCH codes and the dimension of this family of negacyclic BCH codes with large designed distance. The main objective of this paper is to study three subfamilies of this family of negacyclic BCH codes. The dimension and true minimum distance of a subfamily of the q-ary narrow-sense negacyclic BCH codes of length n are determined. The dimension and good lower bounds on the minimum distance of two subfamilies of the q-ary narrow-sense negacyclic BCH codes of length n are presented. The minimum distances of the duals of the q-ary narrow-sense negacyclic BCH codes of length n are also investigated. As will be seen, the three subfamilies of negacyclic BCH codes are sometimes distance-optimal and sometimes have the same parameters as the best linear codes known.
期刊介绍：
Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines.
The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome.
The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas.
Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.
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