Lines in higgledy-piggledy position

TL;DR

This paper investigates the minimum number of lines in projective space meeting each hyperplane in a generator set, establishing tight bounds over different fields and exploring connections to subspace designs.

Contribution

It proves lower bounds on the number of lines meeting hyperplanes in generator sets and shows these bounds are tight over algebraically closed fields, also relating to subspace design parameters.

Findings

At least 1.5d lines if |F| > 1.5d

At least 2d-1 lines if F is algebraically closed

Construction of sets with 2d-1 lines matching bounds

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.