{"title":"利用任意扭曲广义Reed-Solomon码构造最大距离可分码的研究","authors":"Chun'e Zhao;Wenping Ma;Tongjiang Yan;Yuhua Sun","doi":"10.1109/TIT.2025.3563664","DOIUrl":null,"url":null,"abstract":"Maximum distance separable (MDS) codes have significant combinatorial and cryptographic applications due to their certain optimality. Generalized Reed-Solomon (GRS) codes are the most prominent MDS codes. Twisted generalized Reed-Solomon (TGRS) codes may not necessarily be MDS. It is meaningful to study the conditions under which TGRS codes are MDS. In this paper, we study a general class of TGRS (A-TGRS) codes which include all the known special ones. First, we obtain another expression of the inverse of the Vandermonde matrix. Based on this, we further derive an equivalent condition under which an A-TGRS code is MDS. According to this, the A-TGRS MDS codes include nearly all the known related results in the previous literatures. More importantly, we also give three constructions to obtain many other classes of MDS TGRS codes with new parameter matrices. In addition, we present a new method to compute the inverse of the lower triangular Toplitz matrix by a linear feedback shift register, which will be very useful in many research fields.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 7","pages":"5130-5143"},"PeriodicalIF":2.9000,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Research on the Construction of Maximum Distance Separable Codes via Arbitrary Twisted Generalized Reed-Solomon Codes\",\"authors\":\"Chun'e Zhao;Wenping Ma;Tongjiang Yan;Yuhua Sun\",\"doi\":\"10.1109/TIT.2025.3563664\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Maximum distance separable (MDS) codes have significant combinatorial and cryptographic applications due to their certain optimality. Generalized Reed-Solomon (GRS) codes are the most prominent MDS codes. Twisted generalized Reed-Solomon (TGRS) codes may not necessarily be MDS. It is meaningful to study the conditions under which TGRS codes are MDS. In this paper, we study a general class of TGRS (A-TGRS) codes which include all the known special ones. First, we obtain another expression of the inverse of the Vandermonde matrix. Based on this, we further derive an equivalent condition under which an A-TGRS code is MDS. According to this, the A-TGRS MDS codes include nearly all the known related results in the previous literatures. More importantly, we also give three constructions to obtain many other classes of MDS TGRS codes with new parameter matrices. In addition, we present a new method to compute the inverse of the lower triangular Toplitz matrix by a linear feedback shift register, which will be very useful in many research fields.\",\"PeriodicalId\":13494,\"journal\":{\"name\":\"IEEE Transactions on Information Theory\",\"volume\":\"71 7\",\"pages\":\"5130-5143\"},\"PeriodicalIF\":2.9000,\"publicationDate\":\"2025-04-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"IEEE Transactions on Information Theory\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://ieeexplore.ieee.org/document/10975070/\",\"RegionNum\":3,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, INFORMATION SYSTEMS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"IEEE Transactions on Information Theory","FirstCategoryId":"94","ListUrlMain":"https://ieeexplore.ieee.org/document/10975070/","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, INFORMATION SYSTEMS","Score":null,"Total":0}
Research on the Construction of Maximum Distance Separable Codes via Arbitrary Twisted Generalized Reed-Solomon Codes
Maximum distance separable (MDS) codes have significant combinatorial and cryptographic applications due to their certain optimality. Generalized Reed-Solomon (GRS) codes are the most prominent MDS codes. Twisted generalized Reed-Solomon (TGRS) codes may not necessarily be MDS. It is meaningful to study the conditions under which TGRS codes are MDS. In this paper, we study a general class of TGRS (A-TGRS) codes which include all the known special ones. First, we obtain another expression of the inverse of the Vandermonde matrix. Based on this, we further derive an equivalent condition under which an A-TGRS code is MDS. According to this, the A-TGRS MDS codes include nearly all the known related results in the previous literatures. More importantly, we also give three constructions to obtain many other classes of MDS TGRS codes with new parameter matrices. In addition, we present a new method to compute the inverse of the lower triangular Toplitz matrix by a linear feedback shift register, which will be very useful in many research fields.
期刊介绍:
The IEEE Transactions on Information Theory is a journal that publishes theoretical and experimental papers concerned with the transmission, processing, and utilization of information. The boundaries of acceptable subject matter are intentionally not sharply delimited. Rather, it is hoped that as the focus of research activity changes, a flexible policy will permit this Transactions to follow suit. Current appropriate topics are best reflected by recent Tables of Contents; they are summarized in the titles of editorial areas that appear on the inside front cover.