The multicovering radius problem for some types of discrete structures

TL;DR

This paper introduces a generalized multicovering radius concept for permutation codes and codes of perfect matchings, applying probabilistic methods to establish lower bounds and correcting previous errors in related proofs.

Contribution

It extends the multicovering radius concept to new code types and provides probabilistic lower bounds, also correcting earlier proof errors.

Findings

Established lower bounds on multicovering radii for permutation codes

Extended the concept to codes of perfect matchings

Corrected an error in previous covering radius proofs

Abstract

The covering radius problem is a question in coding theory concerned with finding the minimum radius $r$ such that, given a code that is a subset of an underlying metric space, balls of radius $r$ over its code words cover the entire metric space. Klapper introduced a code parameter, called the multicovering radius, which is a generalization of the covering radius. In this paper, we introduce an analogue of the multicovering radius for permutation codes (cf. Keevash and Ku, 2006) and for codes of perfect matchings (cf. Aw and Ku, 2012). We apply probabilistic tools to give some lower bounds on the multicovering radii of these codes. In the process of obtaining these results, we also correct an error in the proof of the lower bound of the covering radius that appeared in Keevash and Ku (2006). We conclude with a discussion of the multicovering radius problem in an even more general context, which offers room for further research.