{"title":"有限环Z15上的线性码","authors":"Wen-sheng Li, Lingyu Wan, Meng-tian Yue, Wei Chen, Xuedong Zhang","doi":"10.4236/alamt.2020.101001","DOIUrl":null,"url":null,"abstract":"In this paper, the structure of the non-chain ring Z15 is studied. The ideals of the ring Z15 are obtained through its non-units and the Lee weights of elements in Z15 are presented. On this basis, by the Chinese Remainder Theorem, we construct a unique expression of an element in Z15. Further, the Gray mapping from Zn15 to Z2n15 is defined and it’s shown to be distance preserved. The relationship between the minimum Lee weight and the minimum Hamming weight of the linear code over the ring Z15 is also obtained and we prove that the Gray map of the linear code over the ring Z15 is also linear.","PeriodicalId":65610,"journal":{"name":"线性代数与矩阵理论研究进展(英文)","volume":"205 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Linear Codes over the Finite Ring Z15\",\"authors\":\"Wen-sheng Li, Lingyu Wan, Meng-tian Yue, Wei Chen, Xuedong Zhang\",\"doi\":\"10.4236/alamt.2020.101001\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, the structure of the non-chain ring Z15 is studied. The ideals of the ring Z15 are obtained through its non-units and the Lee weights of elements in Z15 are presented. On this basis, by the Chinese Remainder Theorem, we construct a unique expression of an element in Z15. Further, the Gray mapping from Zn15 to Z2n15 is defined and it’s shown to be distance preserved. The relationship between the minimum Lee weight and the minimum Hamming weight of the linear code over the ring Z15 is also obtained and we prove that the Gray map of the linear code over the ring Z15 is also linear.\",\"PeriodicalId\":65610,\"journal\":{\"name\":\"线性代数与矩阵理论研究进展(英文)\",\"volume\":\"205 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-04-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"线性代数与矩阵理论研究进展(英文)\",\"FirstCategoryId\":\"1089\",\"ListUrlMain\":\"https://doi.org/10.4236/alamt.2020.101001\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"线性代数与矩阵理论研究进展(英文)","FirstCategoryId":"1089","ListUrlMain":"https://doi.org/10.4236/alamt.2020.101001","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
In this paper, the structure of the non-chain ring Z15 is studied. The ideals of the ring Z15 are obtained through its non-units and the Lee weights of elements in Z15 are presented. On this basis, by the Chinese Remainder Theorem, we construct a unique expression of an element in Z15. Further, the Gray mapping from Zn15 to Z2n15 is defined and it’s shown to be distance preserved. The relationship between the minimum Lee weight and the minimum Hamming weight of the linear code over the ring Z15 is also obtained and we prove that the Gray map of the linear code over the ring Z15 is also linear.
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