On minimum size blocking sets of the outer tangents to a hyperbolic quadric in PG(3,q)

Abstract

Let Q + (3, q) be a hyperbolic quadric in PG(3, q) and T 1 be the set of all lines of PG(3, q) meeting Q + (3, q) in singletons (the so-called outer tangents). If k is the minimum size of a T 1-blocking set in PG(3, q), then we prove that k ≥ q 2 − 1. It is known that there is no T 1-blocking set of size q 2 − 1 for q > 2 even and that there is a unique (up to isomorphism) T 1-blocking set of size 3 for q = 2. For q = 3, we prove as well that there is a unique T 1-blocking set of size 8. Using a computer, we also classify all T 1-blocking sets of size q 2 − 1 for each prime power q ≤ 13. On basis of some structural similarities we are subsequently able to recognize three families of blocking sets whose further study shows that they can be constructed from certain objects related to finite fields (like nice subsets or permutations of the latter). This connection with finite fields allows us to obtain some computer free descriptions.