Evasive subspaces

Let V denote an r-dimensional vector space over F_{q^n}, the finite field of q^n elements. Then V is also an rn-dimension vector space over F_q. An F_q-subspace U of V is (h,k)_q-evasive if it meets the h-dimensional F_{q^n}-subspaces of V in F_q-subspaces of dimension at most k. The (1,1)_q-evasive subspaces are known as scattered and they have been intensively studied in finite geometry, their maximum size has been proved to be [rn/2] when rn is even or n=3. We investigate the maximum size of (h,k)_q-evasive subspaces, study two duality relations among them and provide various constructions. In particular, we present the first examples, for infinitely many values of q, of maximum scattered subspaces when r=3 and n=5. We obtain these examples in characteristics 2, 3 and 5.