On the Sparseness of Certain MRD Codes

Abstract

We determine the proportion of $[3\times 3;3]$-MRD codes over ${\mathbb{F}}_q$ within the space of all $3$-dimensional $3\times 3$-rank-metric codes over the same field. This shows that for these parameters MRD codes are sparse in the sense that the proportion tends to $0$ as $q\to \infty$. This is so far the only parameter case for which MRD codes are known to be sparse. The computation is accomplished by reducing the space of all such rank-metric codes to a space of specific bases and subsequently making use of a result by Menichetti (1973) on 3-dimensional semifields.

References in corpus (1)

• Generalized Twisted Gabidulin Codes