On the Genericity of Maximum Rank Distance and Gabidulin Codes

TL;DR

This paper demonstrates that in large extension fields, randomly chosen codes are likely to be both MRD and non-Gabidulin, highlighting the genericity of these properties in rank-metric code spaces.

Contribution

It proves the genericity of MRD and non-Gabidulin properties for linear rank-metric codes over large extension fields and provides probability bounds based on extension degree.

Findings

Random codes over large fields are typically MRD and non-Gabidulin.

Probability bounds depend on the extension degree m.

Properties are generic over the algebraic closure of the base field.

Abstract

We consider linear rank-metric codes in $\mathbb F_{q^m}^n$. We show that the properties of being MRD (maximum rank distance) and non-Gabidulin are generic over the algebraic closure of the underlying field, which implies that over a large extension field a randomly chosen generator matrix generates an MRD and a non-Gabidulin code with high probability. Moreover, we give upper bounds on the respective probabilities in dependence on the extension degree $m$.