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 ⌊ r n ∕ 2 ⌋ 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.