The Discovery of Simple 7-Designs with Automorphism Group PTL (2, 32)

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 eld 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 ooer a way to solve double coset problems in many important cases. Since the graph iso-morphism 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 rst 7-designs with small 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-(40320 15 + 7; 8; 40320 15). The designs have P ? L(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 nd 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 diophan-tine equations by a f0; 1g-vector. We used the recent improvements by Schnorr of the LLL-algorithm for nding the two solutions to the 3297 system.