Codes for the Asymmetric Damerau–Levenshtein Distance

Codes in the Damerau–Levenshtein metric have been studied recently owing to their applications in DNA-based data storage. In previous work, codes for correcting a single deletion and multiple adjacent transpositions were presented. In this work, we consider a new setting with the asymmetric Damerau–Levenshtein distance where both 0-deletions and adjacent transpositions occur. We first study uniquely-decodable codes and present an optimal code (in the sense that its redundancy is optimal up to a constant additive term) correcting a single 0-deletion or a single adjacent transposition with redundancy log n + 2 bits. Then, we present a construction of codes correcting t 0-deletions and s adjacent transpositions with at most (t + 2s) log n bits of redundancy. Next, we focus on list-decodable codes and construct a list-decodable code with list-size O(n
^min{s+1,t}
) and has at most (max{t, s + 1}) log n bits of redundancy.