Partial covers of PG(n,q) (1210.1002v1)

Abstract: In this paper, we show that a set of q+a hyperplanes, q>13, a<(q-10)/4, that does not cover PG(n,q), does not cover at least q^n-1-aq^n-2 points, and show that this lower bound is sharp. If the number of non- covered points is at most q^n-1, then we show that all non-covered points are contained in one hyperplane. Finally, using a recent result of Blokhuis, Brouwer, and Szonyi [3], we remark that the bound on a for which these results are valid can be improved to a<(q-2)/3 and that this upper bound on a is sharp