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.

中文翻译：

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固定长度 Levenshtein 度量的覆盖代码

摘要

覆盖码或覆盖是一组码字，以这些码字为中心的球的并集覆盖整个空间。通常，问题在于找到覆盖码的最小基数。对于经典汉明度量，固定半径\(R\)的最小覆盖码的大小已知为常数因子。最近对于具有\(R\)插入的代码和具有\(R\)删除的代码获得了类似的结果。在本文中，我们研究固定长度 Levenshtein 度量的空间覆盖，即\(R\)插入和\(R\)删除。对于\(R=1\)和\(2\)，我们证明了覆盖码的最小基数的新下限和上限，它们仅相差一个常数因子。