Lines in higgledy-piggledy position

In this article, we examine sets of lines in PG(d,F) meeting each hyperplane in a generator set of points. We prove that such a set has to contain at least 1.5d lines if the field F has more than 1.5d elements, and at least 2d− 1 lines if the field F is algebraically closed. We show that suitable 2d−1 lines constitute such a set (if |F| ≥ 2d−1), proving that the lower bound is tight over algebraically closed fields. At last, we will see that the strong (s,A) subspace designs constructed by Guruswami and Kopparty [3] have better (smaller) parameter A than one would think at first sight.