We show that small blocking sets in PG(n, q) with respect to hyperplanes intersect every hyperplane in 1 mudulo p points, where q - p(h). The result is then extended to blocking sets with respect to k-dimensional subspaces and at least when p greater than or equal to 2, to intersections with arbitrary subspaces not just hyperplanes. This can also be used to characterize certain non-degenerate blocking sets in higher dimensions. Furthermore we determine the possible sizes of small minimal blocking sets with respect to k-dimensional subspaces. (C) 2001 Academic Press. [References: 16]
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