Codes for Correcting Asymmetric Adjacent Transpositions and Deletions

Owing to the vast applications in DNA-based data storage, Gabrys, Yaakobi, and Milenkovic recently proposed to study codes in the Damerau--Levenshtein metric, where both deletion and adjacent transposition errors occur. In particular, they designed a 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 occur. We present several constructions of the codes correcting these errors in various cases. In particular, we design a code correcting a single deletion, $s^+$ right-shift, and $s^-$ left-shift errors with at most $(1+s)\log (n+s+1)+1$ bits of redundancy where $s=s^{+}+s^{-}$. In addition, we investigate codes correcting $t$ $0$-deletions and $s$ adjacent transpositions with both unique decoding and list-decoding algorithms. Our main contribution here is a construction of a list-decodable code with list-size $O(n^{\min\{s+1,t\}})$ and has at most $(\max \{t,s+1\}) \log n+O(1)$ bits of redundancy. Finally, we provide both non-systematic and systematic codes for correcting $t$ blocks of $0$-deletions with $\ell$-limited-magnitude and $s$ adjacent transpositions.