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, $s^+$ 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, $s^+$ right-shift, and $s^-$ 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 $O(n^{\min\{s+1,t\}})$ and with at most $(\max \{t,s+1\}) \log n+O(1)$ bits of redundancy, where $s=s^{+}+s^{-}$. Finally, we construct both non-systematic and systematic codes for correcting blocks of $0$-deletions with $\ell$-limited-magnitude and $s$ adjacent transpositions.

翻译：Damerau-Levenshtein度量下的编码近年来因在DNA数据存储中的应用而受到广泛研究。具体而言，Gabrys、Yaakobi和Milenkovic（2017）设计了一种长度为$n$的编码，可纠正单个删除和$s$个相邻换位，冗余度最多为$(1+2s)\log n$比特。本文考虑一种新场景，其中可能同时出现非对称相邻换位（亦称右移或左移）和删除。我们针对不同情况提出了纠正这些错误的多种编码构造。特别地，我们设计了一种能纠正单个删除、$s^+$次右移和$s^-$次左移错误的编码，冗余度最多为$(1+s)\log (n+s+1)+1$比特，其中$s=s^{+}+s^{-}$。此外，我们研究了纠正$t$次$0$-删除、$s^+$次右移和$s^-$次左移错误的编码，并给出了唯一译码和列表译码算法。本文的主要贡献在于构建了一种列表可译码的编码，其列表规模为$O(n^{\min\{s+1,t\}})$，冗余度最多为$(\max \{t,s+1\}) \log n+O(1)$比特，其中$s=s^{+}+s^{-}$。最后，我们针对$\ell$限幅幅度和$s$个相邻换位的$0$-删除块纠正，分别构造了非系统码和系统码。