Some Recent Advances on Symmetric, Quasi-Symmetric and Quasi-Multiple Designs

Abstract

Since the advent of design theory that began with constructive results of R.C. Bose and M. Hall, symmetric designs have occupied a very special status. This is due to several reasons, two most important of which are the structural symmetry and the difficulty in constructions of these designs (compared to other classes of designs). The initial part of this exposition will concentrate on new results on symmetric designs. Quasi-symmetric designs are closely related to symmetric designs. These are designs that have (at the most) two block intersection numbers. One can associate a block graph with a quasi-symmetric design and in many cases of interest, this also turns out to be a graph with special properties, and is called a strongly regular graph. One trivial way of constructing quasi-symmetric designs is to take multiple copies of a symmetric design. Since a symmetric design has exactly one block intersection number, the resulting design will have two block intersection numbers. Such quasi-symmetric designs are called improper. Only one example of a proper quasi-multiple quasi-symmetric design seems to be known so far. The question of the existence of a quasi-multiple of a symmetric design is of some importance particularly when the corresponding symmetric design is known not to exist. In that case, obtaining a quasi-multiple with least multiplicity has attracted attention of combinatorialists in the last thirty years.