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 Minimal Multiple Blocking Sets

10.37236/7810 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Anurag Bishnoi ◽  
Sam Mattheus ◽  
Jeroen Schillewaert
Keyword(s):  
Upper Bound ◽  
Prime Power ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Blocking Set ◽  
Blocking Sets ◽  
Incidence Graph ◽  
Finite Projective Spaces ◽  
Power Equality ◽  
Minimal Blocking

We prove that a minimal $t$-fold blocking set in a finite projective plane of order $n$ has cardinality at most \[\frac{1}{2} n\sqrt{4tn - (3t + 1)(t - 1)} + \frac{1}{2} (t - 1)n + t.\] This is the first general upper bound on the size of minimal $t$-fold blocking sets in finite projective planes and it generalizes the classical result of Bruen and Thas on minimal blocking sets. From the proof it directly follows that if equality occurs in this bound then every line intersects the blocking set $S$ in either $t$ points or $\frac{1}{2}(\sqrt{4tn  - (3t + 1)(t - 1)}  + t - 1) + 1$ points. We use this to show that for $n$ a prime power, equality can occur in our bound in exactly one of the following three cases: (a) $t = 1$, $n$ is a square and $S$ is a unital; (b) $t = n - \sqrt{n}$, $n$ is a square and $S$ is the complement of a Baer subplane; (c) $t = n$ and $S$ is equal to the set of all points except one. For a square prime power $q$ and $t \leq \sqrt{q} + 1$, we give a construction of a minimal $t$-fold blocking set $S$ in $\mathrm{PG}(2,q)$ with $|S| = q\sqrt{q} + 1 + (t - 1)(q - \sqrt{q} + 1)$. Furthermore, we obtain an upper bound on the size of minimal blocking sets in symmetric $2$-designs and use it to give new proofs of other known results regarding tangency sets in higher dimensional finite projective spaces. We also discuss further generalizations of our bound. In our proofs we use an incidence bound on combinatorial designs which follows from applying the expander mixing lemma to the incidence graph of these designs.

A Class of Non-Desarguesian Projective Planes

1957 ◽  
Vol 9 ◽  
pp. 378-388 ◽  
Author(s):  
D. R. Hughes
Keyword(s):  
Positive Integer ◽  
Projective Plane ◽  
Collineation Group ◽  
Difference Sets ◽  
Projective Planes ◽  
Desarguesian Projective Plane ◽  
Desarguesian Plane ◽  
Desarguesian Projective Planes

In (7), Veblen and Wedclerburn gave an example of a non-Desarguesian projective plane of order 9; we shall show that this plane is self-dual and can be characterized by a collineation group of order 78, somewhat like the planes associated with difference sets. Furthermore, the technique used in (7) will be generalized and we will construct a new non-Desarguesian plane of order p2n for every positive integer n and every odd prime p.

Three families of multiple blocking sets in Desarguesian projective planes of even order

2012 ◽  
Vol 68 (1-3) ◽  
pp. 49-59
Author(s):  
Anton Betten ◽  
Eun Ju Cheon ◽  
Seon Jeong Kim ◽  
Tatsuya Maruta
Keyword(s):  
Projective Planes ◽  
Blocking Sets ◽  
Desarguesian Projective Planes ◽  
Even Order

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

scholarly journals On minimal blocking sets of the generalized quadrangle $Q(4, q)$

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Miroslava Cimráková ◽  
Veerle Fack
Keyword(s):  
Generalized Quadrangle ◽  
Exhaustive Search ◽  
Blocking Set ◽  
Computer Search ◽  
Blocking Sets ◽  
Parabolic Quadric ◽  
International Audience ◽  
Minimal Blocking

International audience The generalized quadrangle $Q(4,q)$ arising from the parabolic quadric in $PG(4,q)$ always has an ovoid. It is not known whether a minimal blocking set of size smaller than $q^2 + q$ (which is not an ovoid) exists in $Q(4,q)$, $q$ odd. We present results on smallest blocking sets in $Q(4,q)$, $q$ odd, obtained by a computer search. For $q = 5,7,9,11$ we found minimal blocking sets of size $q^2 + q - 2$ and we discuss their structure. By an exhaustive search we excluded the existence of a minimal blocking set of size $q^2 + 3$ in $Q(4,7)$.

Minimal blocking sets in finite planes

1992 ◽  
Vol 43 (1-2) ◽  
pp. 178-187 ◽  
Author(s):  
B. F. Sherman
Keyword(s):  
Blocking Sets ◽  
Finite Planes ◽  
Minimal Blocking

AN ALGORITHM FOR CONSTRUCTING BLOCKING SETS FOR PROJECTIVE PLANES

1992 ◽  
Vol 02 (04) ◽  
pp. 437-442
Author(s):  
RUTH SILVERMAN ◽  
ALAN H. STEIN
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Proper Subset ◽  
Blocking Set ◽  
Blocking Sets ◽  
Unique Point ◽  
Unique Line ◽  
The Family ◽  
Property B

A family of sets is said to have property B(s) if there is a set, referred to as a blocking set, whose intersection with each member of the family is a proper subset of that blocking set and contains fewer than s elements. A finite projective plane is a construction satisfying the two conditions that any two lines meet in a unique point and any two points are on a unique line. In this paper, the authors develop an algorithm of complexity O(n3) for constructing a blocking set for a projective plane of order n.

scholarly journals Daisy structure in Desarguesian projective planes

2003 ◽  
Vol 74 (2) ◽  
pp. 145-154 ◽  
Author(s):  
M. Gabriela Araujo Pardo
Keyword(s):  
Lower Bound ◽  
Projective Plane ◽  
Block Design ◽  
Projective Planes ◽  
Special Structure ◽  
Blocking Set ◽  
Desarguesian Projective Planes

AbstractWe distribute the points and lines ofPG(2, 2n+1) according to a special structure that we call the daisy structure. This distribution is intimately related to a special block design which turns out to be isomorphic toPG(n, 2).We show a blocking set of 3qpoints inPG(2, 2n+1)that intersects each line in at least two points and we apply this to find a lower bound for the heterochromatic number of the projective plane.

Sign in / Sign up

Export Citation Format

Share Document