Blocking sets in PG(2, qn) from cones of PG(2n, q)

Abstract

Let andB be a subset of = PG(2n − 1, q) and a subset of PG(2n, q) respectively, with ⊂ PG(2n, q) andB ⊂ . Denote by K the cone of vertex and baseB and consider the point set B defined byin the André, Bruck-Bose representation of PG(2, q n ) in PG(2n, q) associated to a regular spread S of PG(2n − 1, q). We are interested in finding conditions onB and in order to force the set B to be a minimal blocking set in PG(2, q n ). Our interest is motivated by the following observation. Assume a Property α of the pair ( ,B) forces B to turn out a minimal blocking set. Then one can try to find new classes of minimal blocking sets working with the list of all known pairs ( ,B) with Property α. With this in mind, we deal with the problem in the case is a subspace of PG(2n − 1, q) and B a blocking set in a subspace of PG(2n, q); both in a mutually suitable position. We achieve, in this way, new classes and new sizes of minimal blocking sets in PG(2, q n ), generalizing the main constructions of [14]. For example, for q = 3 h , we get large blocking sets of size q n+2 + 1 (n ≥ 5) and of size greater than q n+2 + q n−6 (n ≥ 6). As an application, a characterization of Buekenhout-Metz unitals in PG(2, q 2k ) is also given.

Blocking Sets in PG(r, q n )

“…In this paper, by using the Barlotti-Cofman representation of a projective space of nonprime order [5], we generalize the cone constructions described in [15,19] and we achieve new classes and new sizes of minimal blocking sets in PG(r, q n ). Choosing ovoids of PG (3, q), Q (4, q) and Q(6, q) as base of our cones, we also obtain some large blocking sets (see Theorems 2,4,[7][8][9][10].…”

“…Example 1 Some families of minimal three-dimensional blocking sets of PG(3, q) of size 2q + 1, 3q + 1 (q > 2), 4q + 1 (q > 2, q even), 3q − 1 (q > 2) and kq + 1 (q > 2, q even, and 2 ≤ k ≤ q − 1) are constructed in [20] and are presented in [15,Sect. 3] as blocking sets of typeB 1 ,B 2 ,B 3 ,B 4 andB 5 , respectively.…”

On MPS construction of blocking sets in projective spaces: A generalization

“…By MPS construction we mean the construction carried out by Mazzocca, Polverino, Storme in [10]: starting from a blocking set in a projective space, one can construct blocking sets in spaces whose order is a power of the original one. The idea of the construction generalizes the planar version of Mazzocca and Polverino in [11]; in this section we follow [6]. We consider the Barlotti-Coffman representation of P G(r, q n 1 ) in P G(nr, q 1 ).…”

Random constructions and density results

“…There are examples in the paper [85] this section was based on. Recently Mazzocca and Polverino [69] and Mazzocca et al [70] constructed new examples.…”