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.
Classification :
51E21, 05B25, 11T06
Mots-clés :
minimal blocking set, Baer subplane, stabiliser of a Baer subplane, Hall plane, André plane, double blocking set, value set of polynomials
Affiliations des auteurs :
Jan De Beule
1
;
Tamás Héger
2
;
Tamás Szőnyi
2
;
Geertrui Van de Voorde
3
1
Vrije Universiteit Brussel
2
Eötvös Loránd University
3
Ghent University
@article{10_37236_5717,
author = {Jan De Beule and Tam\'as H\'eger and Tam\'as Sz\H{o}nyi and Geertrui Van de Voorde},
title = {Blocking and double blocking sets in finite planes},
journal = {The electronic journal of combinatorics},
year = {2016},
volume = {23},
number = {2},
doi = {10.37236/5717},
zbl = {1338.51008},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5717/}
}
TY - JOUR
AU - Jan De Beule
AU - Tamás Héger
AU - Tamás Szőnyi
AU - Geertrui Van de Voorde
TI - Blocking and double blocking sets in finite planes
JO - The electronic journal of combinatorics
PY - 2016
VL - 23
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/5717/
DO - 10.37236/5717
ID - 10_37236_5717
ER -
%0 Journal Article
%A Jan De Beule
%A Tamás Héger
%A Tamás Szőnyi
%A Geertrui Van de Voorde
%T Blocking and double blocking sets in finite planes
%J The electronic journal of combinatorics
%D 2016
%V 23
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/5717/
%R 10.37236/5717
%F 10_37236_5717
Jan De Beule; Tamás Héger; Tamás Szőnyi; Geertrui Van de Voorde. Blocking and double blocking sets in finite planes. The electronic journal of combinatorics, Tome 23 (2016) no. 2. doi: 10.37236/5717
