Beyond Single-Deletion Correcting Codes: Substitutions and Transpositions

TL;DR

This paper advances the design of low-redundancy error-correcting codes capable of handling combined deletions, substitutions, and transpositions, especially relevant for DNA data storage, by constructing efficient codes with near-optimal redundancy.

Contribution

It introduces new linear-time encodable and decodable codes for multiple error types, employing weighted VT sketches and providing simpler proofs and near-optimal redundancy bounds.

Findings

Constructed non-binary codes correcting a single edit with nearly optimal redundancy.

Developed binary list-decodable codes for one deletion and substitution with redundancy close to bounds.

Proved the existence of binary codes correcting one deletion or transposition with near-optimal redundancy.

Abstract

We consider the problem of designing low-redundancy codes in settings where one must correct deletions in conjunction with substitutions or adjacent transpositions; a combination of errors that is usually observed in DNA-based data storage. One of the most basic versions of this problem was settled more than 50 years ago by Levenshtein, or one substitution, with nearly optimal redundancy. However, this approach fails to extend to many simple and natural variations of the binary single-edit error setting. In this work, we make progress on the code design problem above in three such variations: We construct linear-time encodable and decodable length-$n$ non-binary codes correcting a single edit error with nearly optimal redundancy $\log n+O(\log\log n)$, providing an alternative simpler proof of a result by Cai, Chee, Gabrys, Kiah, and Nguyen (IEEE Trans. Inf. Theory 2021). This is achieved by employing what we call weighted VT sketches, a notion that may be of independent interest. We construct linear-time encodable and list-decodable binary codes with list-size $2$ for one deletion and one substitution with redundancy $4\log n+O(\log\log n)$. This matches the existential bound up to an $O(\log\log n)$ additive term. We show the existence of a binary code correcting one deletion or one adjacent transposition with nearly optimal redundancy $\log n+O(\log\log n)$.