On the Maximum Size of Codes Under the Damerau-Levenshtein Metric

TL;DR

This paper establishes theoretical upper bounds on the size of error-correcting codes under the Damerau-Levenshtein metric, which accounts for deletions, insertions, substitutions, and transpositions, with implications for DNA storage systems.

Contribution

It provides the first known upper bounds on code sizes in the Damerau-Levenshtein metric and shows the optimality of a specific code correcting deletions and transpositions.

Findings

Upper bounds for code sizes under the Damerau-Levenshtein metric are established.

The code correcting one deletion and adjacent transpositions by Wang et al. is shown to be nearly optimal.

Results have implications for DNA-based storage error correction.

Abstract

The Damerau-Levenshtein distance between two sequences is the minimum number of operations (deletions, insertions, substitutions, and adjacent transpositions) required to convert one sequence into another. Notwithstanding a long history of this metric, research on error-correcting codes under this distance has remained limited. Recently, motivated by applications in DNA-based storage systems, Gabrys \textit{et al} and Wang \texit{et al} reinvigorated interest in this metric. In their works, some codes correcting both deletions and adjacent transpositions were constructed. However, theoretical upper bounds on code sizes under this metric have not yet been established. This paper seeks to establish upper bounds for code sizes in the Damerau-Levenshtein metric. Our results show that the code correcting one deletion and asymmetric adjacent transpositions proposed by Wang \textit{et al} achieves optimal redundancy up to an additive constant.