  We investigate two fundamental questions intersecting coding theory and
combinatorial geometry, with emphasis on their connections. These are the
problem of computing the asymptotic density of MRD codes in the rank metric,
and the Critical Problem for combinatorial geometries by Crapo and Rota. Using
methods from semifield theory, we derive two lower bounds for the density
function of full-rank, square MRD codes. The first bound is sharp when the
matrix size is a prime number and the underlying field is sufficiently large,
while the second bound applies to the binary field. We then take a new look at
the Critical Problem for combinatorial geometries, approaching it from a
qualitative, often asymptotic, viewpoint. We illustrate the connection between
this very classical problem and that of computing the asymptotic density of MRD
codes. Finally, we study the asymptotic density of some special families of
codes in the rank metric, including the symmetric, alternating and Hermitian
ones. In particular, we show that the optimal codes in these three contexts are
sparse.
