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 (2017) designed a length-n code correcting a single deletion and s adjacent transpositions with at most (1+2 s ) 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 → 10) and at most s − 0-left shifts (i.e., 10 → 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 + 2 s ) 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 b blocks of 0-deletions with l -limited magnitude and s adjacent transpositions with redundancy at most (2( t b +2 s )+1) log( n +1)+ O (1) bits and systematic codes with at most ((2( t b + 2 s ) + 1)(1 + 1/(log( t b l + 4))) log( N + 1) + O (log log N ) redundant bits in an N -length codeword.