{"title":"一类非链环上循环码的结构与等级","authors":"Nikita Jain, Sucheta Dutt, Ranjeet Sehmi","doi":"10.1155/2024/8817721","DOIUrl":null,"url":null,"abstract":"The rings <span><svg height=\"12.1436pt\" style=\"vertical-align:-3.18148pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.96212 24.414 12.1436\" width=\"24.414pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,8.931,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,16.783,0)\"></path></g></svg><span></span><svg height=\"12.1436pt\" style=\"vertical-align:-3.18148pt\" version=\"1.1\" viewbox=\"27.2691838 -8.96212 19.997 12.1436\" width=\"19.997pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,27.319,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,33.221,0)\"><use xlink:href=\"#g113-91\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,42.152,3.132)\"><use xlink:href=\"#g50-53\"></use></g></svg></span> have been classified into chain rings and nonchain rings based on the values of <span><svg height=\"14.7729pt\" style=\"vertical-align:-3.181499pt\" version=\"1.1\" viewbox=\"-0.0498162 -11.5914 21.344 14.7729\" width=\"21.344pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g185-47\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,5.902,-5.741)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,14.48,0)\"></path></g></svg><span></span><svg height=\"14.7729pt\" style=\"vertical-align:-3.181499pt\" version=\"1.1\" viewbox=\"24.9261838 -11.5914 24.414 14.7729\" width=\"24.414pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,24.976,0)\"><use xlink:href=\"#g113-91\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,33.907,3.132)\"><use xlink:href=\"#g50-53\"></use></g><g transform=\"matrix(.013,0,0,-0.013,41.759,0)\"><use xlink:href=\"#g117-36\"></use></g></svg><span></span><span><svg height=\"14.7729pt\" style=\"vertical-align:-3.181499pt\" version=\"1.1\" viewbox=\"52.2461838 -11.5914 20.044 14.7729\" width=\"20.044pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,52.296,0)\"><use xlink:href=\"#g185-47\"></use></g><g transform=\"matrix(.013,0,0,-0.013,58.198,0)\"><use xlink:href=\"#g113-91\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,67.129,3.132)\"><use xlink:href=\"#g50-53\"></use></g></svg>.</span></span> In this paper, the structure of a cyclic code of arbitrary length over the rings <span><svg height=\"12.1436pt\" style=\"vertical-align:-3.18148pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.96212 24.414 12.1436\" width=\"24.414pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-91\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,8.931,3.132)\"><use xlink:href=\"#g50-53\"></use></g><g transform=\"matrix(.013,0,0,-0.013,16.783,0)\"><use xlink:href=\"#g117-36\"></use></g></svg><span></span><svg height=\"12.1436pt\" style=\"vertical-align:-3.18148pt\" version=\"1.1\" viewbox=\"27.2691838 -8.96212 19.997 12.1436\" width=\"19.997pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,27.319,0)\"><use xlink:href=\"#g185-47\"></use></g><g transform=\"matrix(.013,0,0,-0.013,33.221,0)\"><use xlink:href=\"#g113-91\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,42.152,3.132)\"><use xlink:href=\"#g50-53\"></use></g></svg></span> for those values of <svg height=\"11.8239pt\" style=\"vertical-align:-0.2325001pt\" version=\"1.1\" viewbox=\"-0.0498162 -11.5914 10.9702 11.8239\" width=\"10.9702pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g185-47\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,5.902,-5.741)\"><use xlink:href=\"#g50-51\"></use></g></svg> for which these are nonchain rings has been established. A unique form of generators for a cyclic code over these rings has also been obtained. Furthermore, the rank and cardinality of a cyclic code over these rings have been established by finding a minimal spanning set for the code.","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"21 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Structure and Rank of a Cyclic Code over a Class of Nonchain Rings\",\"authors\":\"Nikita Jain, Sucheta Dutt, Ranjeet Sehmi\",\"doi\":\"10.1155/2024/8817721\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The rings <span><svg height=\\\"12.1436pt\\\" style=\\\"vertical-align:-3.18148pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.96212 24.414 12.1436\\\" width=\\\"24.414pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,8.931,3.132)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,16.783,0)\\\"></path></g></svg><span></span><svg height=\\\"12.1436pt\\\" style=\\\"vertical-align:-3.18148pt\\\" version=\\\"1.1\\\" viewbox=\\\"27.2691838 -8.96212 19.997 12.1436\\\" width=\\\"19.997pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,27.319,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,33.221,0)\\\"><use xlink:href=\\\"#g113-91\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,42.152,3.132)\\\"><use xlink:href=\\\"#g50-53\\\"></use></g></svg></span> have been classified into chain rings and nonchain rings based on the values of <span><svg height=\\\"14.7729pt\\\" style=\\\"vertical-align:-3.181499pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -11.5914 21.344 14.7729\\\" width=\\\"21.344pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g185-47\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,5.902,-5.741)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,14.48,0)\\\"></path></g></svg><span></span><svg height=\\\"14.7729pt\\\" style=\\\"vertical-align:-3.181499pt\\\" version=\\\"1.1\\\" viewbox=\\\"24.9261838 -11.5914 24.414 14.7729\\\" width=\\\"24.414pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,24.976,0)\\\"><use xlink:href=\\\"#g113-91\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,33.907,3.132)\\\"><use xlink:href=\\\"#g50-53\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,41.759,0)\\\"><use xlink:href=\\\"#g117-36\\\"></use></g></svg><span></span><span><svg height=\\\"14.7729pt\\\" style=\\\"vertical-align:-3.181499pt\\\" version=\\\"1.1\\\" viewbox=\\\"52.2461838 -11.5914 20.044 14.7729\\\" width=\\\"20.044pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,52.296,0)\\\"><use xlink:href=\\\"#g185-47\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,58.198,0)\\\"><use xlink:href=\\\"#g113-91\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,67.129,3.132)\\\"><use xlink:href=\\\"#g50-53\\\"></use></g></svg>.</span></span> In this paper, the structure of a cyclic code of arbitrary length over the rings <span><svg height=\\\"12.1436pt\\\" style=\\\"vertical-align:-3.18148pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.96212 24.414 12.1436\\\" width=\\\"24.414pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-91\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,8.931,3.132)\\\"><use xlink:href=\\\"#g50-53\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,16.783,0)\\\"><use xlink:href=\\\"#g117-36\\\"></use></g></svg><span></span><svg height=\\\"12.1436pt\\\" style=\\\"vertical-align:-3.18148pt\\\" version=\\\"1.1\\\" viewbox=\\\"27.2691838 -8.96212 19.997 12.1436\\\" width=\\\"19.997pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,27.319,0)\\\"><use xlink:href=\\\"#g185-47\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,33.221,0)\\\"><use xlink:href=\\\"#g113-91\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,42.152,3.132)\\\"><use xlink:href=\\\"#g50-53\\\"></use></g></svg></span> for those values of <svg height=\\\"11.8239pt\\\" style=\\\"vertical-align:-0.2325001pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -11.5914 10.9702 11.8239\\\" width=\\\"10.9702pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g 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A unique form of generators for a cyclic code over these rings has also been obtained. Furthermore, the rank and cardinality of a cyclic code over these rings have been established by finding a minimal spanning set for the code.\",\"PeriodicalId\":54214,\"journal\":{\"name\":\"Journal of Mathematics\",\"volume\":\"21 1\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-01-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1155/2024/8817721\",\"RegionNum\":4,\"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":"Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/8817721","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Structure and Rank of a Cyclic Code over a Class of Nonchain Rings
The rings have been classified into chain rings and nonchain rings based on the values of . In this paper, the structure of a cyclic code of arbitrary length over the rings for those values of for which these are nonchain rings has been established. A unique form of generators for a cyclic code over these rings has also been obtained. Furthermore, the rank and cardinality of a cyclic code over these rings have been established by finding a minimal spanning set for the code.
期刊介绍：
Journal of Mathematics is a broad scope journal that publishes original research articles as well as review articles on all aspects of both pure and applied mathematics. As well as original research, Journal of Mathematics also publishes focused review articles that assess the state of the art, and identify upcoming challenges and promising solutions for the community.
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