Sets with fixed congruence in a steiner system

1996 ◽  
Vol 56 (1-2) ◽  
pp. 131-141
Author(s):  
Sandro Rajola
Keyword(s):  
Steiner System ◽  
Fixed Congruence

scholarly journals Analysis of Some Properties from Construction of Finite Projective Planes of Order 2 and 3

CAUCHY ◽  
2016 ◽  
Vol 4 (3) ◽  
pp. 131
Author(s):  
Vira Hari Krisnawati ◽  
Corina Karim
Keyword(s):  
Automorphism Group ◽  
Projective Plane ◽  
Symmetric Group ◽  
Block Design ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Steiner System ◽  
Incidence Relation ◽  
Finite Projective Planes ◽  
Set Of Points

<p class="abstract"><span lang="IN">In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system <em>S</em>(<em>t</em>, <em>k</em>, <em>v</em>) is a set of <em>v</em> points and <em>k</em> blocks which satisfy that every <em>t</em>-subset of <em>v</em>-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with <em>t</em> = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.</span></p><p class="abstract"><span lang="IN">In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.</span></p>

Direct Product of Derived Steiner Systems Using Inversive Planes

1981 ◽  
Vol 33 (6) ◽  
pp. 1365-1369 ◽  
Author(s):  
K. T. Phelps
Keyword(s):  
Direct Product ◽  
Triple System ◽  
Steiner Triple System ◽  
Steiner System ◽  
Steiner Systems

A Steiner system S(t, k, v) is a pair (P, B) where P is a v-set and B is a collection of k-subsets of P (usually called blocks) such that every t-subset of P is contained in exactly one block of B. As is well known, associated with each point x ∈ P is a S(t � 1, k � 1, v � 1) defined on the set Px = P\{x} with blocksB(x) = {b\{x}|x ∈ b and b ∈ B}.The Steiner system (Px, B(x)) is said to be derived from (P, B) and is called (obviously) a derived Steiner (t – 1, k – 1)-system. Very little is known about derived Steiner systems despite much effort (cf. [11]). It is not even known whether every Steiner triple system is derived.Steiner systems are closely connected to equational classes of algebras (see [7]) for certain values of k.

Finite noncrystallographic groups, 11-vertex equi-edged triangulated clusters and polymorphic transformations in metals

2014 ◽  
Vol 70 (6) ◽  
pp. 616-625 ◽  
Author(s):  
Alexander Talis ◽  
Valentin Kraposhin
Keyword(s):  
Experimental Data ◽  
Polymorphic Transformation ◽  
Steiner System ◽  
Polymorphic Transformations ◽  
Tetrahedral Cluster ◽  
Mathieu Group ◽  
One To One ◽  
The One ◽  
System Mapping

The one-to-one correspondence has been revealed between a set of cosets of the Mathieu groupM11, a set of blocks of the Steiner systemS(4, 5, 11) and 11-vertex equi-edged triangulated clusters. The revealed correspondence provides the structure interpretation of theS(4, 5, 11) system: mapping of the biplane 2-(11, 5, 2) onto the Steiner systemS(4, 5, 11) determines uniquely the 11-vertex tetrahedral cluster, and the automorphisms of theS(4, 5, 11) system determine uniquely transformations of the said 11-vertex tetrahedral cluster. The said transformations correspond to local reconstructions during polymorphic transformations in metals. The proposed symmetry description of polymorphic transformation in metals is consistent with experimental data.

scholarly journals On the construction of the Steiner system S(5, 8, 24)

1977 ◽  
Vol 47 (1) ◽  
pp. 77-79 ◽  
Author(s):  
David R Mason
Keyword(s):  
Steiner System

scholarly journals A Steiner system S (5,6,108)

1994 ◽  
Vol 125 (1-3) ◽  
pp. 183-186 ◽  
Author(s):  
M.J. Grannell ◽  
T.S. Griggs
Keyword(s):  
Steiner System

scholarly journals EXISTENCE OF -ANALOGS OF STEINER SYSTEMS

2016 ◽  
Vol 4 ◽  
Author(s):  
MICHAEL BRAUN ◽  
TUVI ETZION ◽  
PATRIC R. J. ÖSTERGÅRD ◽  
ALEXANDER VARDY ◽  
ALFRED WASSERMANN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Automorphism Groups ◽  
Steiner System ◽  
Dimensional Subspace ◽  
Steiner Systems ◽  
Cover Problem ◽  
Exact Cover

Let $\mathbb{F}_{q}^{n}$ be a vector space of dimension $n$ over the finite field $\mathbb{F}_{q}$. A $q$-analog of a Steiner system (also known as a $q$-Steiner system), denoted ${\mathcal{S}}_{q}(t,\!k,\!n)$, is a set ${\mathcal{S}}$ of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$ such that each $t$-dimensional subspace of $\mathbb{F}_{q}^{n}$ is contained in exactly one element of ${\mathcal{S}}$. Presently, $q$-Steiner systems are known only for $t\,=\,1\!$, and in the trivial cases $t\,=\,k$ and $k\,=\,n$. In this paper, the first nontrivial $q$-Steiner systems with $t\,\geqslant \,2$ are constructed. Specifically, several nonisomorphic $q$-Steiner systems ${\mathcal{S}}_{2}(2,3,13)$ are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of $\text{GL}(13,2)$. This approach leads to an instance of the exact cover problem, which turns out to have many solutions.

scholarly journals Octad Orbits for Certain Subgroups of

2021 ◽  
pp. 155-195
Author(s):  
Peter Rowley ◽  
◽  
Louise Walker ◽  
Keyword(s):  
Steiner System ◽  
Mathieu Group ◽  
Fischer Group

Using Curtis’s MOG [3], we display the orbits and orbit representatives for various subgroups of the Mathieu group acting on the octads of the Steiner system This information is deployed in [8] and [9] to study a graph associated with the largest simple Fischer group.

scholarly journals Laplacian Integral of Particular Steiner System

2021 ◽  
Vol 3 (1) ◽  
pp. 31-32
Author(s):  
Alfi Yusrotis Zakiyyah
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Steiner System

The notion of a hypergraph is motivated by a graph. In graph, every edge contains of two vertices. However, a hypergraph edges contains more than two vertices. In this article use hyperedge to mention edge of hypergraph. A finite projective plane of order n, denoted by  , is a linear intersecting hypergraph. In this research finite projective plane order  is Laplacian integral.

scholarly journals Residual $q$-Fano Planes and Related Structures

10.37236/7107 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Tuvi Etzion ◽  
Niv Hooker
Keyword(s):  
Block Design ◽  
Infinite Family ◽  
Steiner System ◽  
Interesting Case ◽  
Common Belief ◽  
Steiner Systems ◽  
Fano Plane ◽  
Open Problems ◽  
One Step ◽  
Existence Question

One of the most intriguing problems for $q$-analogs of designs, is the existence question of an infinite family of $q$-Steiner systems that are not spreads. In particular the most interesting case is the existence question for the $q$-analog of the Fano plane, known also as the $q$-Fano plane. These questions are in the front line of open problems in block design. There was a common belief and a conjecture that such structures do not exist. Only recently, $q$-Steiner systems were found for one set of parameters. In this paper, a definition for the $q$-analog of the residual design is presented. This new definition is different from previous known definition, but its properties reflect better the $q$-analog properties. The existence of a design with the parameters of the residual $q$-Steiner system in general and the residual $q$-Fano plane in particular are examined. We construct different residual $q$-Fano planes for all $q$, where $q$ is a prime power. The constructed structure is just one step from a construction of a $q$-Fano plane.

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