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 at least 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 have better (smaller) parameter A than one would think at first sight.
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Fancsali et al. (2014) studied this question.