Note on the existence of large minimal blocking sets in galois planes

COMBINATORICA ◽  
1992 ◽  
Vol 12 (2) ◽  
pp. 227-235 ◽  
Author(s):  
Tamás Szőnyi
Keyword(s):  
Blocking Sets ◽  
Minimal Blocking ◽  
Galois Planes

scholarly journals The Lower Bounds of Eight and Fourth Blocking Sets and Existence of Minimal Blocking Sets

2007 ◽  
Vol 19 (3) ◽  
pp. 99-111
Author(s):  
L.Yasin Nada Yassen Kasm Yahya ◽  
Abdul Khalik
Keyword(s):  
Lower Bounds ◽  
Blocking Sets ◽  
Minimal Blocking

Note on disjoint blocking sets in Galois planes

2006 ◽  
Vol 14 (2) ◽  
pp. 149-158 ◽  
Author(s):  
János Barát ◽  
Stefano Marcugini ◽  
Fernanda Pambianco ◽  
Tamás Szőnyi
Keyword(s):  
Blocking Sets ◽  
Galois Planes

On a family of minimal blocking sets of the Hermitian surface

2011 ◽  
Vol 19 (4) ◽  
pp. 313-316 ◽  
Author(s):  
Antonio Cossidente ◽  
Oliver H. King
Keyword(s):  
Blocking Sets ◽  
Hermitian Surface ◽  
Minimal Blocking

Largest minimal blocking sets in PG(2,8)

2003 ◽  
Vol 11 (3) ◽  
pp. 162-169 ◽  
Author(s):  
J. Barát ◽  
S. Innamorati
Keyword(s):  
Blocking Sets ◽  
Minimal Blocking

scholarly journals Blocking and Double Blocking Sets in Finite Planes

10.37236/5717 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jan De Beule ◽  
Tamás Héger ◽  
Tamás Szőnyi ◽  
Geertrui Van de Voorde
Keyword(s):  
Projective Planes ◽  
Blocking Set ◽  
Blocking Sets ◽  
Desarguesian Plane ◽  
Baer Subplanes ◽  
Desarguesian Projective Planes ◽  
Finite Planes ◽  
Value Sets ◽  
Minimal Blocking ◽  
Desarguesian Affine Plane

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.

scholarly journals On the Linearity of Higher-Dimensional Blocking Sets

10.37236/446 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
G. Van De Voorde
Keyword(s):  
Dimensional Space ◽  
Proper Subset ◽  
Blocking Set ◽  
Blocking Sets ◽  
Higher Dimensional ◽  
Linearity Conjecture ◽  
Minimal Blocking

A small minimal $k$-blocking set $B$ in $\mathrm{PG}(n,q)$, $q=p^t$, $p$ prime, is a set of less than $3(q^k+1)/2$ points in $\mathrm{PG}(n,q)$, such that every $(n-k)$-dimensional space contains at least one point of $B$ and such that no proper subset of $B$ satisfies this property. The linearity conjecture states that all small minimal $k$-blocking sets in $\mathrm{PG}(n,q)$ are linear over a subfield $\mathbb{F}_{p^e}$ of $\mathbb{F}_q$. Apart from a few cases, this conjecture is still open. In this paper, we show that to prove the linearity conjecture for $k$-blocking sets in $\mathrm{PG}(n,p^t)$, with exponent $e$ and $p^e\geq 7$, it is sufficient to prove it for one value of $n$ that is at least $2k$. Furthermore, we show that the linearity of small minimal blocking sets in $\mathrm{PG}(2,q)$ implies the linearity of small minimal $k$-blocking sets in $\mathrm{PG}(n,p^t)$, with exponent $e$, with $p^e\geq t/e+11$.

On the non-trivial minimal blocking sets in binary projective spaces

2021 ◽  
Vol 72 ◽  
pp. 101814
Author(s):  
Nanami Bono ◽  
Tatsuya Maruta ◽  
Keisuke Shiromoto ◽  
Kohei Yamada
Keyword(s):  
Projective Spaces ◽  
Blocking Sets ◽  
Minimal Blocking

scholarly journals Constructing Minimal Blocking Sets Using Field Reduction

2015 ◽  
Vol 24 (1) ◽  
pp. 36-52
Author(s):  
Geertrui Van de Voorde
Keyword(s):  
Blocking Sets ◽  
Minimal Blocking

scholarly journals On multiple blocking sets in Galois planes

2007 ◽  
Vol 7 (1) ◽  
pp. 39-53 ◽  
Author(s):  
A Blokhuis ◽  
L Lovász ◽  
L Storme ◽  
T Szőnyi
Keyword(s):  
Pairwise Disjoint ◽  
London Math ◽  
Algebraic Curves ◽  
Blocking Sets ◽  
Baer Subplane ◽  
Baer Subplanes ◽  
Galois Planes

AbstractThis 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.

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