Partial covers of PG(n,q)

TL;DR

This paper investigates the minimal number of points not covered by certain hyperplanes in projective spaces over finite fields, establishing bounds and structural properties of uncovered points, with implications for combinatorial geometry.

Contribution

It provides new bounds on the number of non-covered points and characterizes their geometric configuration, improving previous results with sharp bounds.

Findings

Non-covered points are contained in one hyperplane if their number is at most q^{n-1}.

Lower bound on non-covered points is sharp for given hyperplane sets.

Bound on parameter a can be improved to (q-2)/3, which is also sharp.

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