Spaces of matrices of constant rank and uniform vector bundles

We consider the problem of determining l(r,a) , the maximal dimension of a subspace of axa matrices of rank r. We first review (and reprove) in the language of vector bundles the known results. Then using known facts on uniform vectro bundles we prove some new results and make a conjecture. We determine l(r,a) for every r when a <= 10, proving in this case our conjecture