{"title":"Linear Codes Correcting Repeated Bursts Equipped with Homogeneous Distance","authors":"","doi":"10.1007/s00224-024-10166-y","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>The homogeneous weight (metric) is useful in the construction of codes over a ring of integers <span> <span>\\(\\mathbb {Z}_{p^l}\\)</span> </span> (<em>p</em> prime and <span> <span>\\(l \\ge 1\\)</span> </span> an integer). It becomes Hamming weight when the ring is taken to be a finite field and becomes Lee weight when the ring is taken to be <span> <span>\\(\\mathbb {Z}_{4}\\)</span> </span>. This paper presents homogeneous weight distribution and total homogeneous weight of burst and repeated burst errors in the code space of <em>n</em>-tuples over <span> <span>\\(\\mathbb {Z}_{p^l}\\)</span> </span>. Necessary and sufficient conditions for existence of an (<em>n</em>, <em>k</em>) linear code over <span> <span>\\(\\mathbb {Z}_{p^l}\\)</span> </span> correcting the error patterns with respect to the homogeneous weight are derived.</p>","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"16 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Computing Systems","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1007/s00224-024-10166-y","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
The homogeneous weight (metric) is useful in the construction of codes over a ring of integers \(\mathbb {Z}_{p^l}\) (p prime and \(l \ge 1\) an integer). It becomes Hamming weight when the ring is taken to be a finite field and becomes Lee weight when the ring is taken to be \(\mathbb {Z}_{4}\). This paper presents homogeneous weight distribution and total homogeneous weight of burst and repeated burst errors in the code space of n-tuples over \(\mathbb {Z}_{p^l}\). Necessary and sufficient conditions for existence of an (n, k) linear code over \(\mathbb {Z}_{p^l}\) correcting the error patterns with respect to the homogeneous weight are derived.
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