Codes for Correcting Asymmetric Adjacent Transpositions and Deletions

TL;DR

This paper introduces new error-correcting codes for sequences affected by asymmetric adjacent transpositions and deletions, with improved redundancy bounds and decoding algorithms for various error scenarios, relevant for DNA data storage.

Contribution

It presents novel constructions of codes that correct asymmetric transpositions and deletions, including single and multiple errors, with efficient decoding and redundancy guarantees.

Findings

Designed codes correcting single deletion and multiple asymmetric transpositions with low redundancy.

Developed list-decoding algorithms for multiple deletions and transpositions with bounded list size.

Constructed systematic and non-systematic codes for block deletions with limited-magnitude errors.

Abstract

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.