External flats to varieties in

Let M = Mn,n(q) be the space of n × n matrices over the finite field GF(q), q a prime power. For each t ≤ n a subspace Φ(t) of dimension n2 − tn is constructed which contains no nonzero matrix of rank strictly less than t + 1, and it is demonstrated that these subspaces are maximal with respect to this property.