Covering codes for the fixed length Levenshtein metric

Abstract

A covering code, or a covering, is a set of codewords such that the union of balls centered at these codewords covers the whole space. As a rule, the problem consists in nding the minimum of a covering code. For the classical Hamming metric, the size of the smallest covering code of a xed radius R is known up to a constant factor. A similar result has recently been obtained for codes with R insertions and codes with R deletions. In the present paper we study coverings of a space for the xed length Levenshtein metric, i.e., for R insertions and R deletions. For R = 1, 2 we prove new lower and upper bounds on the minimum cardinality of a covering code, which di er by a constant factor only.