Large affine spaces of matrices with rank bounded below

Let K be an arbitrary (commutative) field with at least three elements, and let n, p and r be positive integers with . In a recent work [11], , we have proved that an affine subspace of Mn,p(K) containing only matrices of rank greater than or equal to r must have a codimension greater than or equal to . Here, we classify, up to equivalence, these subspaces of minimal codimension . This uses our recent classification [10] of the affine subspaces of Mr(K) contained in GLr(K) which have the maximal dimension .