Rank-metric codes over arbitrary fields: Bounds and constructions
Alessandro Neri, Ferdinando Zullo
TL;DR
This survey synthesizes the theory of rank-metric codes over arbitrary fields, focusing on bounds, constructions, and linearity structures. It develops the DG construction for fields with cyclic Galois extensions, links MRD codes to systems and scattered/evasive subspaces, and analyzes bounds over algebraically closed fields and the real numbers. The work highlights the duality and equivalence of codes, presents puncturing techniques to derive non-square MRD codes, and outlines open problems and promising directions, including an MRD existence conjecture and extensions to general field settings. The results connect algebraic, geometric, and topological methods, with implications for code design over diverse field families and for understanding rank-metric codes in nonfinite contexts.
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.