Identifiers for MRD-codes

TL;DR

This paper investigates invariants of MRD codes over finite fields, providing new characterizations of generalized twisted Gabidulin codes and highlighting the scarcity of known families despite their abundance.

Contribution

It introduces new invariants for MRD codes and offers a novel characterization of generalized twisted Gabidulin codes, expanding understanding of their structure.

Findings

Almost all rank distance codes are MRD when field extensions are large enough.

Few families of MRD codes are known up to equivalence.

New invariants help distinguish and characterize MRD code families.

Abstract

For any admissible value of the parameters $n$ and $k$ there exist $[n,k]$-Maximum Rank distance ${\mathbb F}_q$-linear codes. Indeed, it can be shown that if field extensions large enough are considered, almost all rank distance codes are MRD. On the other hand, very few families up to equivalence of such codes are currently known. In the present paper we study some invariants of MRD codes and evaluate their value for the known families, providing a new characterization of generalized twisted Gabidulin codes.