Lines in higgledy-piggledy position

Abstract: 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| > or = 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 have better (smaller) parameter A than one would think at first sight.