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

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 $Q(4,7)$.

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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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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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Minimal Multiple Blocking Sets

The Electronic Journal of Combinatorics ◽

10.37236/7810 ◽

2018 ◽

Vol 25 (4) ◽

Cited By ~ 1

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.

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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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minimal blocking set of size (30) in PG (2,19) plane

JOURNAL OF EDUCATION AND SCIENCE ◽

10.33899/edusj.2012.59202 ◽

2012 ◽

Vol 25 (3) ◽

pp. 191-205

Author(s):

Amani Al-Salim

Keyword(s):

Blocking Set ◽

Minimal Blocking

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Blocking sets for cycles and paths designs

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm181108008b ◽

2020 ◽

Vol 14 (1) ◽

pp. 183-197

Author(s):

Paola Bonacini ◽

Lucia Marino

Keyword(s):

Blocking Set ◽

Blocking Sets

In this paper, we study blocking sets for C4, P3 and P5-designs. In the case of C4-designs and P3-designs we determine the cases in which the blocking sets have the largest possible range of cardinalities. These designs are called largely blocked. Moreover, a blocking set T for a G-design is called perfect if in any block the number of edges between elements of T and elements in the complement is equal to a constant. In this paper, we consider perfect blocking sets for C4-designs and P5-designs.

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Embedding the complement of a minimal blocking set in a projective plane

Discrete Mathematics ◽

10.1016/0012-365x(84)90099-2 ◽

1984 ◽

Vol 52 (1) ◽

pp. 1-5

Author(s):

Lynn Margaret Batten

Keyword(s):

Projective Plane ◽

Blocking Set ◽

Minimal Blocking

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Blocking Sets and Skew Subspaces of Projective Space

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-048-8 ◽

1980 ◽

Vol 32 (3) ◽

pp. 628-630 ◽

Cited By ~ 23

Author(s):

Aiden A. Bruen

Keyword(s):

Projective Space ◽

Projective Plane ◽

Projective Spaces ◽

Higher Dimensions ◽

Blocking Set ◽

Blocking Sets ◽

Partial Spreads ◽

Dimension One

In what follows, a theorem on blocking sets is generalized to higher dimensions. The result is then used to study maximal partial spreads of odd-dimensional projective spaces.Notation. The number of elements in a set X is denoted by |X|. Those elements in a set A which are not in the set Bare denoted by A — B. In a projective space Σ = PG(n, q) of dimension n over the field GF(q) of order q, ┌d(Ωd, Λd, etc.) will mean a subspace of dimension d. A hyperplane of Σ is a subspace of dimension n — 1, that is, of co-dimension one.A blocking set in a projective plane π is a subset S of the points of π such that each line of π contains at least one point in S and at least one point not in S. The following result is shown in [1], [2].

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