Generation of Steiner Quadruple Systems

A block design with v points and a set of blocks B where each block is a b-subset of v, such that each point is contained in exactly r-blocks & each distinct k point is contained in exactly -blocks, known as a t- (v, k,) -design which plays an important role in design theory. A Steiner system is a special type of t- (v, k,) design with =1 & k=t+1. 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 2n vertices, where n-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 n- 3, resolvable & non-disjoint. By extending the program for Steiner triple systems blocks of STS (2n-1) -design number of blocks, incidence matrices, and complete graphs with 2n-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.