Covering Properties of Sum-Rank Metric Codes

TL;DR

This paper explores the covering properties of sum-rank metric codes, establishing bounds and relations to other metrics, thereby advancing the theoretical understanding of these codes.

Contribution

It provides new bounds and relations for the minimum size of sum-rank metric codes with a given covering radius, enriching the theoretical framework.

Findings

Derived lower and upper bounds for sum-rank covering codes

Established relations between sum-rank, rank, and Hamming metrics

Enhanced understanding of the covering properties of sum-rank codes

Abstract

The sum-rank metric can be seen as a generalization of both, the rank and the Hamming metric. It is well known that sum-rank metric codes outperform rank metric codes in terms of the required field size to construct maximum distance separable codes (i.e., the codes achieving the Singleton bound in the corresponding metric). In this work, we investigate the covering property of sum-rank metric codes to enrich the theory of sum-rank metric codes. We intend to answer the question: what is the minimum cardinality of a code given a sum-rank covering radius? We show the relations of this quantity between different metrics and provide several lower and upper bounds for sum-rank metric codes.