Covering Codes using Insertions or Deletions

TL;DR

This paper investigates covering codes based on insertions and deletions, establishing new bounds and demonstrating the existence of codes with densities close to theoretical limits, extending classical covering code theory beyond Hamming metrics.

Contribution

It introduces new lower and upper bounds for covering codes under insertion and deletion metrics, and shows the existence of near-optimal codes with asymptotic densities matching Hamming covering codes.

Findings

New sphere-covering lower bounds for insertion/deletion codes

Existence of codes with densities within a factor O(R log R) of lower bounds

Upper bounds with optimal dependence on word length

Abstract

A covering code is a set of codewords with the property that the union of balls, suitably defined, around these codewords covers an entire space. Generally, the goal is to find the covering code with the minimum size codebook. While most prior work on covering codes has focused on the Hamming metric, we consider the problem of designing covering codes defined in terms of either insertions or deletions. First, we provide new sphere-covering lower bounds on the minimum possible size of such codes. Then, we provide new existential upper bounds on the size of optimal covering codes for a single insertion or a single deletion that are tight up to a constant factor. Finally, we derive improved upper bounds for covering codes using $R\geq 2$ insertions or deletions. We prove that codes exist with density that is only a factor $O(R \log R)$ larger than the lower bounds for all fixed~$R$. In particular, our upper bounds have an optimal dependence on the word length, and we achieve asymptotic density matching the best known bounds for Hamming distance covering codes.