Optimal Codes Correcting a Substring Edit

Abstract

The substring edit error replaces a substring $\boldsymbol {u}$ of $\boldsymbol {x}$ with another string $\boldsymbol {v}$ , where the lengths of $\boldsymbol {u}$ and $\boldsymbol {v}$ are bounded by a given constant k. It encompasses localized insertions, deletions, and substitutions within a window. Codes correcting one substring edit have redundancy at least $\log n+k$ . In this paper, we construct codes correcting one substring edit with redundancy $\log n+O_{k}(\log \log n)$ , which is almost optimal. We also study the average-case document-exchange problem under one substring edit and construct a hash with an expected length of approximately $2\log n+O_{k}(\log \log n)$ for any iid distribution for the documents.