Bounds and Constructions for Insertion and Deletion Codes

TL;DR

This paper investigates theoretical bounds and practical constructions of insertion and deletion codes, revealing limitations of existing bounds and proposing new code constructions that approach optimal insdel distances.

Contribution

It establishes that nontrivial insdel codes cannot achieve the insdel-metric Singleton bound and introduces new constructions of codes nearing this bound.

Findings

Nontrivial insdel codes do not reach the insdel-metric Singleton bound.

Reed-Solomon codes have an insdel distance upper bounded by 2n-4k+4.

Constructed nonlinear and Reed-Solomon codes approach the insdel-metric Singleton bound.

Abstract

The present paper mainly studies limits and constructions of insertion and deletion (insdel for short) codes. The paper can be divided into two parts. The first part focuses on various bounds, while the second part concentrates on constructions of insdel codes. Although the insdel-metric Singleton bound has been derived before, it is still unknown if there are any nontrivial codes achieving this bound. Our first result shows that any nontrivial insdel codes do not achieve the insdel-metric Singleton bound. The second bound shows that every $[n,k]$ Reed-Solomon code has insdel distance upper bounded by $2n-4k+4$ and it is known in literature that an $[n,k]$ Reed-Solomon code can have insdel distance $2n-4k+4$ as long as the field size is sufficiently large. The third bound shows a trade-off between insdel distance and code alphabet size for codes achieving the Hamming-metric Singleton bound. In the second part of the paper, we first provide a non-explicit construction of nonlinear codes that can approach the insdel-metric Singleton bound arbitrarily when the code alphabet size is sufficiently large. The second construction gives two-dimensional Reed-Solomon codes of length $n$ and insdel distance $2n-4$ with field size $q=O(n^5)$.