Combinatorial alphabet-dependent bounds for insdel codes

TL;DR

This paper introduces new combinatorial bounds for insdel codes, improving existing limits by employing linear programming and hypergraph matchings, with implications for DNA storage and biological data correction.

Contribution

It provides novel upper and lower bounds on insdel code sizes, including a sphere-packing LP bound and hypergraph-based bounds, extending the theoretical understanding of these codes.

Findings

Improved sphere-packing upper bound for large q and d

New lower bounds based on hypergraph matchings

Refined GV-type bounds applicable to large q and d

Abstract

Error-correcting codes resilient to synchronization errors such as insertions and deletions are known as insdel codes. Due to their important applications in DNA storage and computational biology, insdel codes have recently become a focal point of research in coding theory. In this paper, we present several new combinatorial upper and lower bounds on the maximum size of $q$-ary insdel codes. Our main upper bound is a sphere-packing bound obtained by solving a linear programming (LP) problem. It improves upon previous results for cases when the distance $d$ or the alphabet size $q$ is large. Our first lower bound is derived from a connection between insdel codes and matchings in special hypergraphs. This lower bound, together with our upper bound, shows that for fixed block length $n$ and edit distance $d$, when $q$ is sufficiently large, the maximum size of insdel codes is $ \frac{q^{n-\frac{d}{2}+1}}{{n\choose \frac{d}{2}-1}}(1 \pm o(1))$. The second lower bound refines Alon et al.'s recent logarithmic improvement on Levenshtein's GV-type bound and extends its applicability to large $q$ and $d$.