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.

翻译：由于在基于DNA的数据存储中的广泛应用，Gabrys、Yaakobi和Milenkovic近期提出研究Damerau–Levenshtein度量下的编码，其中同时出现删除错误和相邻转位错误。他们特别设计了一种编码，能够纠正单个删除错误和$s$个相邻转位错误，冗余度最多为$(1+2s)\log n$比特。本文考虑一种新场景，其中同时出现非对称相邻转位（也称为右移或左移）和删除错误。我们针对不同情况提出了多种纠正此类错误的编码构造。特别地，我们设计了一种编码，能够纠正单个删除错误、$s^+$个右移错误和$s^-$个左移错误，冗余度最多为$(1+s)\log (n+s+1)+1$比特，其中$s=s^{+}+s^{-}$。此外，我们研究了能够纠正$t$个0-删除错误和$s$个相邻转位错误的编码，并给出了唯一译码和列表译码算法。这里的主要贡献是构造了一种列表可译编码，其列表大小为$O(n^{\min\{s+1,t\}})$，冗余度最多为$(\max \{t,s+1\}) \log n+O(1)$比特。最后，我们针对纠正$t$块$\ell$限幅0-删除错误和$s$个相邻转位错误，分别给出了非系统码和系统码构造。