ENUMERATION OF STEINER TRIPLE SYSTEM
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Date
2018
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Department of Mathematics and Statistics Sam Higginbottom University of Agriculture, Technology and Sciences Allahabad-211007 (U. P.) India
Abstract
Steiner triple system, denoted by STS v , is a pair S , T consisting of a set S with V
elements, and a set T consisting of Triples of S (Called blocks) such that every pair of
elements of S appear together in a unique triple of T .
The higher order Steiner system is very complex in nature. Various methodologies can be
used to construct and enumerate the STS for order of 23, 25, 27 and 29. The various
combinatory theory and design theory can be used for the enumeration of the desired
STS .
Here, in this work, we have presented a detailed method of the enumeration methodology
and construction of the STS systems.
Detailed numerical analysis has been done for the enumeration of the STS of order 23,
25, 27 and 29. The algorithm discussed will here generate the total possible STS
combinations. Finally, construction methodology has also been presented.
Graphical construction has been discussed with the reference to the kirkman systems. The
properties of the STS system equations have been discussed. The various theorems have
been represented using the order 9 subsystems. Results have shown the steps of the
enumeration and combinations of various pairs showing the properties of STS of the
order 23, 25, 27 and 29.
Steiner triple systems STSs with subsystems of order 7 are classified. The Steiner
systems for order 19, 21 and 23 has been classified at first then 25, 27, and 29. We need
to find the total classification of the STS systems. In a Steiner triple system of order v ,
STS v , a set of three lines intersecting pair wise in three distinct points is called a
triangle.
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A set of lines containing no triangle is called triangle-free. The minimum number of
triangle-free sets required to partition the lines of a Steiner triple system S , is called the
triangle chromatic index of S .
A method of constructing Steiner triple systems has been proposed. The basic theory that
contains definition, lemmas of decomposable of some complete graphs has been
described. A theorem concerning construction of Steiner triple systems has been proved.
The entire procedure of constructing Steiner triple systems of order 23, 25, 27 and 29 has
been presented.
A Steiner triple system of order v , is a set of 3-element subsets, called blocks, of a v -set
of points, such that every pair of points occurs in exactly one block. For a STS v ,
standard counting arguments show that each point must occur in exactly r v 1 / 2
blocks and that the triple system consists of exactly b v v 1 / 6 blocks where r and b
are integers we get necessary conditions for the existence of a STS v
For v 3, a STS v exists if and only if either v 1 mod 6 or v 3 mod 6 . Two STS
are isomorphic if there exists a bijection between the point sets that maps blocks onto
blocks; such a bijection is an isomorphism. An auto-morphism of a triple system is an
isomorphism of the triple system onto itself. The auto-morphism group of the triple
system consists of all of its automorphisms. The number of pair wise non-isomorphic
STS v is denoted by N v .
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Description
ph.d . thesis
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