A computer approach to the enumeration of block designs which are invariant with respect to a prescribed permutation group

We present a new approach to the construction of simple block designs. Using the computer package DISCRETA, we start with the construction of block designs which are invariant with respect to some prescribed group of automorphisms. Therefore, one applies the method of Kramer and Mesner which means that one has to solve systems of diophantine equations to get the designs. DISCRETA has proven its usefulness for the construction of t-designs with large t and a prescribed group of automorphisms. In this paper, we generalize its application to the construction of designs on small point sets with no prescribed group. It is interesting to see that – to some extent – one can apply DISCRETA even in that extreme case. But the solutions to the Kramer-Mesner system are not necessarily distinct from the point of view of the designs: Some designs may be obtained repeatedly and it is the question to find them and to put away duplicates. We present a new way for doing this by looking at the minimal subgroups of Sv, the symmetric group on all points. This approach avoids going through the whole lattice of subgroups which can be quite big even for symmetric groups of small degree. Using a combination of the computer packages GAP, GRAPE, nauty and COCO it is even possible to determine the full automorphism groups for all the isomorphism classes of designs. Small designs (with v ≤ 9) are revisited and numerous new designs with the parameters (11, 55, 20, 4, 6), (11, 110, 40, 4, 12), (13, 78, 30, 5, 10), (14, 91, 26, 4, 6), (14, 91, 39, 6, 15), (15, 42, 14, 5, 4), (15, 105, 35, 5, 10) and (17, 68, 16, 4, 3) are discovered.