scholarly journals Blocking sets for cycles and paths designs

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.

Blocking Sets and Skew Subspaces of Projective Space

1980 ◽  
Vol 32 (3) ◽  
pp. 628-630 ◽  
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].

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

Blocking Sets in Projective Spaces

1978 ◽  
Vol 30 (4) ◽  
pp. 856-862
Author(s):  
Gary L. Ebert
Keyword(s):  
Game Theory ◽  
Projective Plane ◽  
Projective Spaces ◽  
Blocking Set ◽  
Blocking Sets ◽  
Partial Spreads ◽  
Image Position

Blocking sets in projective spaces have been of interest for quite some time, having applications to game theory (see [6; 7]) as well as finite nets and partial spreads (see [5]). In [4] Bruen showed that if B is a blocking set in a projective plane of order n, then

scholarly journals The Optimal Size of {b, t} - Blocking Set When t = 3,4 by Intersection the Tangent in PG (2, q)

2019 ◽  
Vol 13 (7) ◽  
pp. 15
Author(s):  
Shaymaa Haleem Ibrahim ◽  
Nada Yassen Kasm
Keyword(s):  
Blocking Set ◽  
Optimal Size ◽  
Blocking Sets ◽  
Projection Plane

In this research, we have been able to construct a triple blocking set of optimal size - {4q, 3} Based on the theorem (1.4.7) (Maruta, 2017, pp. 1-47).Without improving the minimum constraint of the projection level PG (2, q) We have also been able to develop the theorem (2.3.2) to construct quadratic blocking set of optimal size {5q + 1,4} - After we have engineered a quadratic blocking set of an optimal size for the projection plane PG (2,1.3) In the example (2.3.1).In general, we were able to conclude theorems (2.3.3) and (2.3.4) for construct engineered blocking sets with an optimal size when t = 3,4.

scholarly journals A New Lower Bound for the Size of an Affine Blocking Set

10.37236/7827 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Maarten De Boeck ◽  
Geertrui Van de Voorde
Keyword(s):  
Lower Bound ◽  
Affine Plane ◽  
Blocking Set ◽  
Blocking Sets ◽  
Affine Planes ◽  
Set Of Points

A blocking set in an affine plane is a set of points $B$ such that every line contains at least one point of $B$. The best known lower bound for blocking sets in arbitrary (non-desarguesian) affine planes was derived in the 1980's by Bruen and Silverman. In this note, we improve on this result by showing that a blocking set of an affine plane of order $q$, $q\geqslant 25$, contains at least $q+\lfloor\sqrt{q}\rfloor+3$ points.

scholarly journals The Possibility of Applying Rumen Research at the Projective Plane PG (2,17)

2019 ◽  
Vol 13 (8) ◽  
pp. 150
Author(s):  
Shaymaa Haleem Ibrahim ◽  
Nada Yassen Kasm
Keyword(s):  
Projective Plane ◽  
Theoretical Method ◽  
Galois Field ◽  
Blocking Set ◽  
Blocking Sets ◽  
Projection Plane ◽  
Set Of Points

One of the main objectives of this research is to use a new theoretical method to find arcs and Blocking sets. This method includes the deletion of a set of points from some lines under certain conditions explained in a paragraph 2.In this paper we were able to improve the minimum constraint of the (256,16) – arc in the projection plane PG(2,17).Thus , we obtained a new {50,2}-blocking set for size Less than 3q , and according to the theorem (1.3.1),we obtained the linear 257,3,24117    code, theorem( 2.1.1 ) giving some examples on arcs of the Galois field GF(q);q=17."

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 $Q(4,7)$.

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.

Blocking Sets in SQS(2v)

1994 ◽  
Vol 3 (1) ◽  
pp. 77-86 ◽  
Author(s):  
Mario Gionfriddo ◽  
Salvatore Milici ◽  
Zsolt Tuza
Keyword(s):  
Blocking Set ◽  
Blocking Sets ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Element Subset ◽  
Steiner Quadruple System ◽  
Quadruple System ◽  
Doubling Construction ◽  
Necessary And Sufficient ◽  
Element Set

A Steiner quadruple system SQS(v) of order v is a family ℬ of 4-element subsets of a v-element set V such that each 3-element subset of V is contained in precisely one B ∈ ℬ. We prove that if T ∩ B ≠ ø for all B ∈ ℬ (i.e., if T is a transversal), then |T| ≥ v/2, and if T is a transversal of cardinality exactly v/2, then V \ T is a transversal as well (i.e., T is a blocking set). Also, in respect of the so-called ‘doubling construction’ that produces SQS(2v) from two copies of SQS(v), we give a necessary and sufficient condition for this operation to yield a Steiner quadruple system with blocking sets.

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