Blocking and Double Blocking Sets in Finite Planes

In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order $q^2$ of size $q^2+2q+2$ admitting $1-$, $2-$, $3-$, $4-$, $(q+1)-$ and $(q+2)-$secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order $q^2$ of size at most $4q^2/3+5q/3$, which is considerably smaller than $2q^2-1$, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order $q^2$.We also consider particular André planes of order $q$, where $q$ is a power of the prime $p$, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in $1$ mod $p$ points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results.

On multiple blocking sets in Galois planes

This article continues the study of multiple blocking sets in PG(2,q). In [A. Blokhuis, L. Storme, T. Szőnyi, Lacunary polynomials, multiple blocking sets and Baer subplanes.J. London Math. Soc. (2)60(1999), 321–332. MR1724814 (2000j:05025) Zbl 0940.51007], using lacunary polynomials, it was proven thatt-fold blocking sets of PG(2,q),qsquare,t<q¼/2, of size smaller thant(q+ 1) +cqq⅔, withcq= 2−⅓whenqis a power of 2 or 3 andcq= 1 otherwise, contain the union oftpairwise disjoint Baer subplanes whent≥ 2, or a line or a Baer subplane whent= 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 at(modp) result on smallt-fold blocking sets of PG(2,q=pn),pprime,n≥ 1.