A small basis for four-line configurations in steiner triple systems. Dedicated to the memory of gemma holly griggs (original) (raw)

Steiner Triple Systems Of Order 25

2017

The higher order Steiner system is very complex in nature. Various methodologies can be used to construct and enumerate the STS for order of 25. The various combinatory theory and design theory can be used for the enumeration of the desired STS. Here, in this paper, 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 25. 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 25. -------------------------------------------------...

" Construction Methodology and enumeration of the Steiner Triple Systems of Order 27 "

This paper describes the construction and enumeration of the Steiner Triple system of large order systems. It describes the methodology use to generate the triplets for the systems of order 27. We have presented the methodology of enumerated permutation of the diagonal matrices. Enumeration of the total Steiner Triple systems of order 27 has been deduced. In the proposed methodology, combinatorics has been applied and problems for the enumeration and calculation have also been discussed.

Generation of Steiner Quadruple Systems

International Journal of Science and Research (IJSR), 2014

A block design with points and a set of blocks where each block is a-subset of , such that each point is contained in exactly-blocks & each distinct point is contained in exactly-blocks, known as a − , ,-design which plays an important role in design theory. A Steiner system is a special type of − (, ,) design with = & = +. Among these steiner systems, steiner quadruple systems (SQS) and Steiner Triple Systems(STS) are the designs that are widely used in constructing designs. In this work, we present an effective automated method of finding SQS design of 2 n vertices, where ∈ ℤ, with the help of STS. We begin with a set of blocks of a known STS, and the binary representation of all those blocks was constructed. Then, a MATLAB program was used to find the blocks of a SQS which related to the SQS that we have chosen. The next step was to find the corresponding incidence matrix for the design obtained in the first step and another separate program was designed to obtain the incidence matrix. Finally, with the help of this incidence matrix, a new program was implemented to obtain a complete graph which corresponds to the SQS obtained above. These blocks have several properties such that triply transitive, automorphism-free, heterogeneous for ≥ 3, resolvable & non-disjoint. By extending the program for Steiner triple systems blocks of STS(2 n-1)-design number of blocks, incidence matrices, and complete graphs with 2 n-1 number of vertices were obtained as another result. These Steiner quadruple systems and Steiner triple systems can be used in fields of communication, cryptography, and networking.

Configurations and trades in Steiner triple systems

The main result of this paper is the determination of all pairwise non- isomorphic trade sets of volume at most 10 which can appear in Steiner triple systems. We also enumerate partial Steiner triple systems having at most 10 blocks as well as configurations with no points of degree 1 and tradeable configurations having at most 12 blocks. AMS classification: 05B07 Keywords: Steiner triple system; Configurations; Trades.

On the Szamkolowicz-Doyen Classification of Steiner Triple Systems

Proceedings of the London Mathematical Society, 1977

1. Introduction A Steiner triple system (8, stf) is a finite set 8 together with a family s/ of triples (subsets of cardinality 3) such that each pair of elements of S is contained in exactly one triple. The elements of 8 are called points, the elements of s/ are called lines, and three points which are not on a line are a triangle. The cardinal of 8, which we denote by 18 |, is the order of the Steiner triple system (S,s/). It is well known [8] that a necessary and sufficient condition for the existence of a Steiner triple system of order n > 0 is that n = 1 or 3 (mod 6). Trivial Steiner triple systems occur when S = stf -0, when 8 consists of one point and s& = 0, and when 8 consists of three points and $2 consists of one line. A subsystem of a Steiner triple system (S, stf) is a Steiner triple system (8', &4') such that 8' <= 8 and s#' <= s0. It is easy to see that if (8 V J^) , ..., (8 r ,s0 r ) are subsystems of a Steiner triple system then so is (8 x n... n8 r ,s^n... ns/ r ). It follows that given a set P s S of points, there is a unique minimal subsystem containing P. This subsystem is said to be generated by P.

Towards a large set of Steiner quadruple systems

SIAM Journal on Discrete Mathematics, 1991

SIAM i-LUST. Math, Й 1991 Society for Industrial and Applied Mathematics Vol.4, No. 2, pp. 182-195, May 19!) I 004 TOWARDS A LARGE SET OF STEINER QUADRUPLE SYSTEMS* TUVI ETZIONf and ALAN HARTMANf Abstract. Let D(u) be the number of pairwise disjoint ...

Properties of the Steiner Triple Systems of Order 19

The Electronic Journal of Combinatorics, 2010

Properties of the 11$\,$084$\,$874$\,$829 Steiner triple systems of order 19 are examined. In particular, there is exactly one 5-sparse, but no 6-sparse, STS(19); there is exactly one uniform STS(19); there are exactly two STS(19) with no almost parallel classes; all STS(19) have chromatic number 3; all have chromatic index 10, except for 4$\,$075 designs with chromatic index 11 and two with chromatic index 12; all are 3-resolvable; and there are exactly two 3-existentially closed STS(19).

Once more about 80 Steiner triple systems on 15 points

Journal of Statistical Planning and Inference, 1998

Subsets of a v-set are in one-to-one correspondence with vertices of a v-dimensional unit cube, a Delaunay polytope of the lattice Zv. All vertices of the same cardinality k generate a (v−1)-dimensional root lattice Av−1 and are vertices of the Delaunay polytope P(v,k) of the lattice Av−1. Hence k-blocks of a t−(v,k,λ) design, being identified with vertices of P(v,k), generate

Linearly Derived Steiner Triple Systems

Designs, Codes and Cryptography - DCC, 1998

We attach a graph to every Steiner triple system. The chromatic number of this graph is related to the possibility of extending the triple system to a quadruple system. For example, the triple systems with chromatic number one are precisely the classical systems of points and lines of a projective geometry over the two-element field, the Hall triple systems have chromatic number three (and, as is well-known, are extendable) and all Steiner triple systems whose graph has chromatic number two are extendable. We also give a configurational characterization of the Hall triple systems in terms of mitres.

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