The Use of Blocking Sets in Galois Geometries and in Related Research Areas

Abstract

Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems. Definitions and introductory resultsA set B of points of PG(n, q) is called a k-blocking set if every (n − k)-dimensional subspace of PG(n, q) has a non-empty intersection with B, and a set B of points of PG(n, q) is called a t-fold k-blocking set if every (n − k)-dimensional subspace contains at least t points of B. In PG(2, q), the 1-blocking sets are simply called blocking sets. Two subspaces Σ 1 and Σ 2 of PG(n, q) of dimension k and n − k always have a non-empty intersection, hence a set B containing a k-dimensional subspace is called a trivial k-blocking set. Moreover, a k-blocking set B is called minimal if it is minimal with respect to the containment relation, i.e. B \ {P } is not a k-blocking set for every P ∈ B, and it is called small if |B| < 3(q k +1) 2 . The blocking sets of PG(2, q) have been extensively studied in the last years and there are plenty of results about characterizations (of small ones) and about the bounds on the size. A trivial blocking set of PG(2, q) is a set of points containing a line. The first examples of non-trivial blocking sets of small size of PG(2, q) have been given in the following: Theorem 1. In PG(2, q), q odd, there exists a projective triangle of side q+3 2 that is a minimal blocking set of size[14]. In PG(2, q), q even, there exists a projective triad of side q+2 2 that is a minimal blocking set of size 3q+2 2[40].This motivates the choice of the boundfor the size of a small blocking set in PG(2, q) and then generalized asfor the k-blocking sets of PG(n, q).