Rank-metric codes over arbitrary fields: Bounds and constructions

TL;DR

This paper surveys the theoretical foundations, bounds, and constructions of rank-metric codes over various fields, highlighting their mathematical properties, limitations, and potential for future research beyond finite fields.

Contribution

It extends the study of rank-metric codes to arbitrary fields, discusses bounds and constructions, and explores their mathematical connections and open problems.

Findings

Singleton-like bounds are sharp over finite fields.

Constructions of MRD codes over cyclic Galois extensions are presented.

Results over algebraically closed fields and real numbers are reviewed.

Abstract

Rank-metric codes, defined as sets of matrices over a finite field with the rank distance, have gained significant attention due to their applications in network coding and connections to diverse mathematical areas. Initially studied by Delsarte in 1978 and later rediscovered by Gabidulin, these codes have become a central topic in coding theory. This paper surveys the development and mathematical foundations, in particular, regarding bounds and constructions of rank-metric codes, emphasizing their extension beyond finite fields to more general settings. We examine Singleton-like bounds on code parameters, demonstrating their sharpness in finite field cases and contrasting this with contexts where the bounds are not tight. Furthermore, we discuss constructions of Maximum Rank Distance (MRD) codes over fields with cyclic Galois extensions and the relationship between linear rank-metric codes with systems and evasive subspaces. The paper also reviews results for algebraically closed fields and real numbers, previously appearing in the context of topology and measure theory. We conclude by proposing future research directions, including conjectures on MRD code existence and the exploration of rank-metric codes over various field extensions.