Lines in Higgledy-Piggledy Arrangement

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 b1.5dc lines if the field F has at least b1.5dc 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 have better (smaller) parameter A than one would think at first sight.