International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 Volume 4 Issue 12, December 2015 www.ijsr.net Licensed Under Creative Commons Attribution CC BY Generation of Steiner Quadruple Systems H. M. I. C. Herath 1 , A. A. I. Perera 2 , A. A. C. A. Jayathilake 3 1, 2, 3 Department of Mathematics, Faculty of Science, University of Peradeniya Abstract: 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. Keywords: Block designs, Steiner quadruple system, Steiner triple system 1. Introduction A block design is a design with points and a set of blocks ℬ where each block contains subsets with points called - subset, such that each point is contained in exactly -blocks &any pair of points is contained in exactly λ-blocks, where r, λ are constants. A t −v, k, λ-design is a design with parameters v, k, λwhere (X, ℬ) is a pair with-point set &the collection of k subsets of X, called blocks with the property that any t- points are contained in exactly λ blocks & > ≥ A − (, , ) design with λ =1 & = +1 is called a Steiner System which was first introduced by the Swiss Mathematician Jacob Steiner(Anderson & Honkala, 1997). Steiner quadruple system is a pair (X, ℬ) where is a finite set and is a collection of 4-subsets (called quadruple or blocks with 4 elements in each) such that every 3-subsets of is contained in exactly one quadruple ofℬ.In other words, a Steiner quadruple system is a Steiner system with = 3 & =4.It has been shown that ≡ 2 4( 6) is the necessary condition for the existence of Steiner quadruple system. (Linder & Rosa, 1977, p. 148) There are various types of Steiner quadruple systems. Among them, cyclic SQS, transitive SQS, automorphism- free SQS, heterogeneous SQS, resolvable SQS & disjoint SQS are some prominent types of SQS(Linder & Rosa, 1977, pp. 154-173). A SQS() is cyclic if it admits an automorphism consisting of single cycle of length , where is called a cyclic automorphism. Also, ≡ 2 10( 24)can be identified as the necessary condition for theexistence of cyclic SQS.Further, it is important to note that, there are no cyclic SQS with order 8, 14 & 16.The next order of thecyclic SQS is 20 and the following figure shows the 15 base quadruples (Linder & Rosa, 1977, pp. 155-157). , + 1, + 3, +4 , + 1, + 2, + 11 { , + 1, + 5, + 16} , + 2, + 6, +8, + 2, + 4, + 12 {, + 3, + 6, + 13} , + 3, + 9, + 14, + 1, + 6, +7 {, + 1, + 9, + 12} , + 1, + 8, + 13, + 2, + 7, +9 {, + 2, + 5, + 17} , + 3, + 7, + 16, + 4, + 8, + 15 {, + 5, + 10, + 15} mod(20). If (X, ℬ) is a cyclic SQS, then the quadruples of ℬcan bepartitioned into orbits under the action of the cyclic group generated by . Each orbit of quadruples is completely determined by any one of its quadruples, and the set ℬ is determined by a collection of quadruple called base quadruples containing one quadruple from each orbit. (Linder & Rosa, 1977, pp. 154-155) SQS whose automorphism group acts transitively on the elements is called transitive SQS(Linder & Rosa, 1977, p. 157). Cyclic Steiner quadruple systems are transitive. Any three points in affine geometryAG(n, 2) determines a plane of AG(n, 2) and this plane contains just one more point of AG(n, 2) (Linder & Rosa, 1977, p. 157). So in SQS(2 n ) design, any three points in a block is uniquely determined the fourth according to the affine geometry. Therefore, the automorphism group of SQS(2 n ) design is triply transitive. Automorphism group of STS(2 n ) which is isomorphic to the design of points and lines of affine space AG(n,3) over GF(3) is also triply transitive. Using that design, sharply triply transitive SQS can be constructed. Another interesting class of transitive SQS (flag-transitiveSQS) can be constructed using STS with prime power order ≡ 7(mod 12). A SQS is automorphism-free, if it admits no non trivialautomorphism. If Steiner quadruplesystem is an automorphism-free then ≥ 16. This is an example of automorphism-free SQS(16)(Linder & Rosa, 1977, p. 159). Paper ID: NOV152246 1551