On the genericity of maximum rank distance and Gabidulin codes

We consider linear rank-metric codes in \({\mathbb {F}}_{q^m}^n\). We show that the properties of being maximum rank distance (MRD) 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.