Codes for Correcting Asymmetric Adjacent Transpositions and Deletions

Abstract: Codes in the Damerau--Levenshtein metric have been extensively studied recently owing to their applications in DNA-based data storage. In particular, Gabrys, Yaakobi, and Milenkovic (2017) designed a length-$n$ code correcting a single deletion and $s$ adjacent transpositions with at most $(1+2s)\log n$ bits of redundancy. In this work, we consider a new setting where both asymmetric adjacent transpositions (also known as right-shifts or left-shifts) and deletions may occur. We present several constructions of the codes correcting these errors in various cases. In particular, we design a code correcting a single deletion, $s<sup>+$ right-shift, and $s<sup>-$ left-shift errors with at most $(1+s)\log (n+s+1)+1$ bits of redundancy where $s=s<sup>{+}+s<sup>{-}$. In addition, we investigate codes correcting $t$ $0$-deletions, $s<sup>+$ right-shift, and $s<sup>-$ left-shift errors with both uniquely-decoding and list-decoding algorithms. Our main contribution here is the construction of a list-decodable code with list size $O(n<sup>{\min{s+1,t}})$ and with at most $(\max {t,s+1}) \log n+O(1)$ bits of redundancy, where $s=s<sup>{+}+s<sup>{-}$. Finally, we construct both non-systematic and systematic codes for correcting blocks of $0$-deletions with $\ell$-limited-magnitude and $s$ adjacent transpositions.