The discovery of simple 7-designs with automorphism group PΓL(2, 32)

Abstract

A computer package is being developed at Bayreuth for the generation and investigation of discrete structures. The package is a C and C++ class library of powerful algorithms endowed with graphical interface modules. Standard applications can be run automatically whereas research projects mostly require small C or C++ programs. The basic philosophy behind the system is to transform problems into standard problems of e.g. group theory, graph theory, linear algebra, graphics, or databases and then to use highly specialized routines from that field to tackle the problems. The transformations required often follow the same principles especially in the case of generation and isomorphism testing.

We therefore explain some of this background.

We relate orbit problems to double cosets and we offer a way to solve double coset problems in many important cases. Since the graph isomorphism problem is equivalent to a certain double coset problem, no polynomial algorithm can be expected to work in the general case. But the reduction techniques used still allow to solve problems of an interesting size. As an example we explain how the 7-designs in the title were found. The two simple 7-designs with parameters 7-(33, 8, 10) and 7-(33, 8, 16) are presented in this paper. To the best of our knowledge they are the first 7-designs with small λ and small number of blocks ever found. Teirlinck [19] had shown previously that non trivial t-designs without repeated blocks exist for all t. The smallest parameters for the case t=7 are 7-(40320¹⁵ + 7,8,40320¹⁵).

The designs have PΓL(2, 32) as automorphism group, and they are constructed from the Kramer-Mesner method [7]. This group had previously been used by [13] in order to find simple 6-designs. The presentation of our results is compatible with that earlier publication.

The Kramer-Mesner method requires to solve a system of linear diophantine equations by a {0, 1}-vector. We used the recent improvements by Schnorr of the LLL-algorithm for finding the two solutions to the 32 x 97 system.

The authors thank C. Praeger for pointing out reference [12] to us. We also thank the referees for helpful suggestions for a detailed presentation of the designs.