On the Genericity of Maximum Rank Distance and Gabidulin Codes

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 $m$.

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• On the number of inequivalent Gabidulin codes
• Nuclei and automorphism groups of generalized twisted Gabidulin codes
• Densities of Codes of Various Linearity Degrees in Translation-Invariant Metric Spaces
• Constructing MRD codes by switching
• A geometric invariant of linear rank-metric codes