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

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].

Partial Spreads and Replaceable Nets

1971 ◽  
Vol 23 (3) ◽  
pp. 381-391 ◽  
Author(s):  
A. Bruen
Keyword(s):  
Projective Plane ◽  
Blocking Set ◽  
Partial Spreads

A blocking set S in a projective plane Π is a subset of the points of Π such that every line of Π contains at least one point of S and at least one point not in S. In previous papers [5; 6], we have shown that if Π is finite of order n, then n + √n + 1 ≦ |S| ≦ n2 – √n (see [6, Theorem 3.9]), where |S| stands for the number of points of S. This work is concerned with some applications of the above result to nets and partial spreads, and with some examples of partial spreads which give rise to unimbeddable nets of small deficiency.In the next section we re-prove a well known result of Bruck which states that if N is a replaceable net of order n and degree k then k ≧ √n + 1, and show how this bound can be improved if n = 7, 8, or 11.

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."

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 Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

10.37236/2582 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Tamás Héger ◽  
Marcella Takáts
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
The Other ◽  
Blocking Set ◽  
Metric Dimension ◽  
Desarguesian Projective Plane ◽  
Resolving Set ◽  
Incidence Graph ◽  
Vertex Set

In a graph $\Gamma=(V,E)$ a vertex $v$ is resolved by a vertex-set $S=\{v_1,\ldots,v_n\}$ if its (ordered) distance list with respect to $S$, $(d(v,v_1),\ldots,d(v,v_n))$, is unique. A set $A\subset V$ is resolved by $S$ if all its elements are resolved by $S$. $S$ is a resolving set in $\Gamma$ if it resolves $V$. The metric dimension of $\Gamma$ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving set is a set of vertices in one of the vertex classes that resolves the other class.We show that the metric dimension of the incidence graph of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in PG$(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of PG$(2,q)$, $|S|\geq \min \{2q+q/4-3, \tau_2-2\}$ holds. In particular, if $q\geq9$ is a square, then the smallest semi-resolving set in PG$(2,q)$ has size $2q+2\sqrt{q}$. As a corollary, we get that a blocking semioval in PG$(2, q)$, $q\geq 4$, has at least $9q/4-3$ points. A corrigendum was added to this paper on March 3, 2017.

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.

The BP-Coaction for Projective Spaces

1978 ◽  
Vol 30 (01) ◽  
pp. 45-53 ◽  
Author(s):  
Donald M. Davis
Keyword(s):  
Hopf Algebra ◽  
Spectral Sequence ◽  
Projective Spaces ◽  
Homology Theory ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
New Information ◽  
Image Position

The Brown-Peterson spectrum BP has been used recently to establish some new information about the stable homotopy groups of spheres [9; 11]. The best results have been achieved by using the associated homology theory BP* ( ), the Hopf algebra BP*(BP), and the Adams-Novikov spectral sequence

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