On multiple blocking sets in Galois planes

Abstract

This article continues the study of multiple blocking sets in PG(2, q). In [3], using lacunary polynomials, it was proven that t-fold blocking sets of PG(2, q), q square, t < q 1/4 /2, of size smaller than t(q + 1) + c q q 2/3 , with c q = 2 −1/3 when q is a power of 2 or 3 and c q = 1 otherwise, contain the union of t pairwise disjoint Baer subplanes when t ≥ 2, or a line or a Baer subplane when t = 1. We now combine the method of lacunary polynomials with the use of algebraic curves to improve the known characterization results on multiple blocking sets and to prove a t (mod p) result on small t-fold blocking sets of PG(2, q = p n ), p prime, n ≥ 1.

The 2‐Blocking Number and the Upper Chromatic Number of PG(2,q)

“…Result 3.13 (Blokhuis, Lovász, Storme, Szőnyi [7]). Let B be a minimal t-fold blocking set in PG(2, q), q = p h , p prime, h ≥ 1, |B| < tq + (q + 3)/2.…”

“…We may assume that P = (x 1 , y 1 ). Let H(B, M ) be the Rédei polynomial of S \ {(∞)}: Since S is a t-fold blocking set, every pair (b, m) produces at least t factors vanishing in H; thus by [6] (or see [7]), H(B, M ) can be written in the form…”

Linear sets in finite projective spaces

“…The results contained in [65,64,69,14,77,13] provide some evidence that the above conjecture holds true.…”

A characterization of multiple (n – k)-blocking sets in projective spaces of square order

“…(This multiple blocking set is a linear point set.) The 1 (mod p) result in PG(2, q), q = p h , p prime, was extended by Blokhuis et al to a t (mod p) result on small minimal t-fold blocking sets in PG(2, q) [7]. Definition 1.10 A t-fold blocking set of PG(2, q) is called small when it has less than (t + 1/2)(q + 1) points.…”

“…Theorem 1.11 (Blokhuis et al [7]) Let B be a small minimal t-fold blocking set in PG(2, q), q = p h , p prime, h ≥ 1. Then B intersects every line in t (mod p) points.…”