On the sparseness of certain linear MRD codes

We determine the proportion of linear [ 3 × 3 ; 3 ] -MRD codes over F q within the space of all 3-dimensional 3 × 3 -rank-metric codes over the same field. This shows that for these parameters linear MRD codes are sparse in the sense that the proportion tends to 0 as q → ∞ . 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.