Minimal blocking sets in finite planes

B. F. Sherman

doi:10.1007/bf01245952

Blocking and Double Blocking Sets in Finite Planes

The Electronic Journal of Combinatorics ◽

10.37236/5717 ◽

2016 ◽

Vol 23 (2) ◽

Cited By ~ 1

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.

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

JOURNAL OF EDUCATION AND SCIENCE ◽

10.33899/edusj.2007.51324 ◽

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

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On a family of minimal blocking sets of the Hermitian surface

Journal of Combinatorial Designs ◽

10.1002/jcd.20279 ◽

2011 ◽

Vol 19 (4) ◽

pp. 313-316 ◽

Cited By ~ 1

Author(s):

Antonio Cossidente ◽

Oliver H. King

Keyword(s):

Blocking Sets ◽

Hermitian Surface ◽

Minimal Blocking

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Largest minimal blocking sets in PG(2,8)

Journal of Combinatorial Designs ◽

10.1002/jcd.10035 ◽

2003 ◽

Vol 11 (3) ◽

pp. 162-169 ◽

Cited By ~ 1

Author(s):

J. Barát ◽

S. Innamorati

Keyword(s):

Blocking Sets ◽

Minimal Blocking

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On the Linearity of Higher-Dimensional Blocking Sets

The Electronic Journal of Combinatorics ◽

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 $p^e\geq t/e+11$.

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