Small blocking sets in higher dimensions

We show that small blocking sets in PG(n, q) with respect to hyperplanes in tersect every hyperplane in 1 mudulo p points, where q - p(h). The result i s then extended to blocking sets with respect to k-dimensional subspaces an d at least when p greater than or equal to 2, to intersections with arbitra ry subspaces not just hyperplanes. This can also be used to characterize ce rtain non-degenerate blocking sets in higher dimensions. Furthermore we det ermine the possible sizes of small minimal blocking sets with respect to k- dimensional subspaces. (C) 2001 Academic Press.