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 r n-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 L r n / 2 RIGHT FLOOR when r n 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.