Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points

The main result of this paper is that point sets of PG(n, q (3)), q = p (h) , p a parts per thousand yen 7 prime, of size less than 3(q (3(n-k)) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p (3)), p a parts per thousand yen 7 prime, of size less than 3(p (3(n-k)) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q (3)) which meet every (n - 2)-space in 1 modulo q points.