{"title":"用邻域搜索算法构造极值$ \\mathbb{Z}_{4} $-码","authors":"Dean Crnković, Matteo Mravić, Sanja Rukavina","doi":"10.3934/amc.2023039","DOIUrl":null,"url":null,"abstract":"In this paper, we present a method for constructing extremal $ \\mathbb{Z}_{4} $-codes based on random neighborhood search. This method is used to find new extremal Type Ⅰ and Type Ⅱ $ \\mathbb{Z}_{4} $-codes of lengths 32 and 40. For the length 32, at least 182 new Type Ⅱ extremal $ \\mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ k\\in\\left\\{9,10,12,13,14,15,16\\right\\} $ are constructed. In addition, we obtained at least 762 new extremal Type Ⅰ $ \\mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ k\\in\\left\\{7,9,10,12,13,14,15,16\\right\\} $. For the length 40, constructed extremal $ \\mathbb{Z}_{4} $-codes are of types $ 4^{k}2^{40-2k} $, $ k\\in\\left\\{7,10,11,15,16\\right\\} $. There are at least 40 new Type Ⅱ extremal $ \\mathbb{Z}_{4} $-codes, and at least 4144 new Type Ⅰ extremal $ \\mathbb{Z}_{4} $-codes.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"40 1","pages":"0"},"PeriodicalIF":0.7000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Construction of extremal $ \\\\mathbb{Z}_{4} $-codes using a neighborhood search algorithm\",\"authors\":\"Dean Crnković, Matteo Mravić, Sanja Rukavina\",\"doi\":\"10.3934/amc.2023039\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we present a method for constructing extremal $ \\\\mathbb{Z}_{4} $-codes based on random neighborhood search. This method is used to find new extremal Type Ⅰ and Type Ⅱ $ \\\\mathbb{Z}_{4} $-codes of lengths 32 and 40. For the length 32, at least 182 new Type Ⅱ extremal $ \\\\mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ k\\\\in\\\\left\\\\{9,10,12,13,14,15,16\\\\right\\\\} $ are constructed. In addition, we obtained at least 762 new extremal Type Ⅰ $ \\\\mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ k\\\\in\\\\left\\\\{7,9,10,12,13,14,15,16\\\\right\\\\} $. For the length 40, constructed extremal $ \\\\mathbb{Z}_{4} $-codes are of types $ 4^{k}2^{40-2k} $, $ k\\\\in\\\\left\\\\{7,10,11,15,16\\\\right\\\\} $. There are at least 40 new Type Ⅱ extremal $ \\\\mathbb{Z}_{4} $-codes, and at least 4144 new Type Ⅰ extremal $ \\\\mathbb{Z}_{4} $-codes.\",\"PeriodicalId\":50859,\"journal\":{\"name\":\"Advances in Mathematics of Communications\",\"volume\":\"40 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Mathematics of Communications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3934/amc.2023039\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"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":"Advances in Mathematics of Communications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/amc.2023039","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Construction of extremal $ \mathbb{Z}_{4} $-codes using a neighborhood search algorithm
In this paper, we present a method for constructing extremal $ \mathbb{Z}_{4} $-codes based on random neighborhood search. This method is used to find new extremal Type Ⅰ and Type Ⅱ $ \mathbb{Z}_{4} $-codes of lengths 32 and 40. For the length 32, at least 182 new Type Ⅱ extremal $ \mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ k\in\left\{9,10,12,13,14,15,16\right\} $ are constructed. In addition, we obtained at least 762 new extremal Type Ⅰ $ \mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ k\in\left\{7,9,10,12,13,14,15,16\right\} $. For the length 40, constructed extremal $ \mathbb{Z}_{4} $-codes are of types $ 4^{k}2^{40-2k} $, $ k\in\left\{7,10,11,15,16\right\} $. There are at least 40 new Type Ⅱ extremal $ \mathbb{Z}_{4} $-codes, and at least 4144 new Type Ⅰ extremal $ \mathbb{Z}_{4} $-codes.
期刊介绍:
Advances in Mathematics of Communications (AMC) publishes original research papers of the highest quality in all areas of mathematics and computer science which are relevant to applications in communications technology. For this reason, submissions from many areas of mathematics are invited, provided these show a high level of originality, new techniques, an innovative approach, novel methodologies, or otherwise a high level of depth and sophistication. Any work that does not conform to these standards will be rejected.
Areas covered include coding theory, cryptology, combinatorics, finite geometry, algebra and number theory, but are not restricted to these. This journal also aims to cover the algorithmic and computational aspects of these disciplines. Hence, all mathematics and computer science contributions of appropriate depth and relevance to the above mentioned applications in communications technology are welcome.
More detailed indication of the journal''s scope is given by the subject interests of the members of the board of editors.