{"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":null,"pages":null},"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.