{"title":"Covering Codes for the Fixed Length Levenshtein Metric","authors":"","doi":"10.1134/s0032946023020023","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>A covering code, or a covering, is a set of codewords such that the union of balls centered at these codewords covers the entire space. As a rule, the problem consists in finding the minimum cardinality of a covering code. For the classical Hamming metric, the size of the smallest covering code of a fixed radius <span> <span>\\(R\\)</span> </span> is known up to a constant factor. A similar result has recently been obtained for codes with <span> <span>\\(R\\)</span> </span> insertions and for codes with <span> <span>\\(R\\)</span> </span> deletions. In the present paper we study coverings of a space for the fixed length Levenshtein metric, i.e., for <span> <span>\\(R\\)</span> </span> insertions and <span> <span>\\(R\\)</span> </span> deletions. For <span> <span>\\(R=1\\)</span> </span> and <span> <span>\\(2\\)</span> </span>, we prove new lower and upper bounds on the minimum cardinality of a covering code, which differ by a constant factor only.</p>","PeriodicalId":54581,"journal":{"name":"Problems of Information Transmission","volume":"89 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Problems of Information Transmission","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1134/s0032946023020023","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
A covering code, or a covering, is a set of codewords such that the union of balls centered at these codewords covers the entire space. As a rule, the problem consists in finding the minimum cardinality of a covering code. For the classical Hamming metric, the size of the smallest covering code of a fixed radius \(R\) is known up to a constant factor. A similar result has recently been obtained for codes with \(R\) insertions and for codes with \(R\) deletions. In the present paper we study coverings of a space for the fixed length Levenshtein metric, i.e., for \(R\) insertions and \(R\) deletions. For \(R=1\) and \(2\), we prove new lower and upper bounds on the minimum cardinality of a covering code, which differ by a constant factor only.
期刊介绍：
Problems of Information Transmission is of interest to researcher in all fields concerned with the research and development of communication systems. This quarterly journal features coverage of statistical information theory; coding theory and techniques; noisy channels; error detection and correction; signal detection, extraction, and analysis; analysis of communication networks; optimal processing and routing; the theory of random processes; and bionics.
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