Enumeration of strongly regular designs

יום א', 25/03/2012 - 14:00

Coherent configurations (CC's), in the sense of D.G. Higman, play a central role in algebraic graph theory. They capture some of the combinatorial properties of permutation groups. However in contrast to other combinatorial structures, no catalogue of such configurations of small order exists. Our long term goal is to fill this gap.

We can classify coherent configurations according to the number of fibers (a combinatorial analogue of orbits of permutation groups). CC's with one fiber are actually association schemes, and for this type of structure a catalogue exists, see http://kissme.shinshu-u.ac.jp/as/.
It is natural to consider CC's with two fibers next.

One particular case of two-fiber CC's is equivalent to strongly regular designs (SRD's), a class of block designs introduced by Higman in 1988. While Higman originally was interested only in structures with primitive point and block graphs, we consider also the imprimitive case. We will
describe our approach to construct all small SRD's and will consider a list of feasible parameter sets that we generated. Then we will give two examples: A classical object (Reye's configuration, discovered in 1882) in a new disguise, and a new object, namely the smallest non-Schurian
SRD with 8 points and 12 blocks.

This is a joint project with Mikhail Klin.