We consider linear rank-metric codes in F n q m . 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.

In this work we develop a geometric approach to the study of rank metric codes. Using this method, we introduce a simpler definition for generalized rank weight of linear codes. We give a complete classification of constant rank weight code and we give their generalized rank weights. *

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The purpose of this paper is to introduce the notion of sum-matroid and to link it to the theory of sum-rank metric codes. Sum-matroids generalize both the notions of matroid and q-matroid. We show how generalized weights can be defined for them and establish a duality for these weights analogous to Wei's one for generalized Hamming weights of linear codes, and more generally, for matroids proved by Britz et al. We associate a sum-matroid to a sum-rank metric code and the corresponding results of Martínez-Peñas for sum-rank metric codes are derived as a consequence.