Binary Codes for Correcting Asymmetric Adjacent Transpositions and Deletions

Codes in the Damerau-Levenshtein metric have received some attention by the research community recently owing to their applications in DNA-based data storage. In particular, Gabrys, Yaakobi, and Milenkovic 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 deletions and asymmetric adjacent transpositions may occur. For asymmetric transpositions, at most $s^{+}0$ -right shifts (i.e., $01 \rightarrow 10$ ) and at most $s^ - 0$ -left shifts (i.e., $10 \rightarrow 01$ ) may occur. We present several constructions of the binary 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 uniquely-decodable codes correcting $t~0$ -deletions and s adjacent transpositions with at most $(t+2s)\log n+o(\log n)$ bits of redundancy. Then, we study the code for correcting $t~0$ -deletions, $s^{+}$ right-shift, and $s^ - $ left-shift errors with list-decoding algorithms. Our main contribution here is the construction of a list-decodable code with list size $O(n^{s})$ and with at most $(\max \{t,s+1\}) \log n+O(1)$ bits of redundancy, where $s=s^{+}+s^ - $ . We construct non-systematic codes for correcting $t_{\mathrm {b}}$ blocks of 0-deletions with $\ell $ -limited magnitude and s adjacent transpositions with redundancy at most $(2(t_{\mathrm {b}}+2s)+1)\log (n+1)+O(1)$ bits and systematic codes with at most $((2(t_{\mathrm {b}}+2s)+1)(1+1/(\log (t_{\mathrm {b}}\ell +4)))\log (N+1)+O(\log \log N)$ redundant bits in an N-length codeword.