Block Designs With Prescribed Automorphism Group

Bernd Schmalz

Séminaire lotharingien de combinatoire, Tome 27 (1991)

Résumé

Kramer and Mesner showed that the t-designs admitting a given automorphism group A are 0-1 solutions x of a system of equations

M^A_t,k . x = (\lambda,...,\lambda)^t.

Based on this approach we present an algorithm for the construction of a system of representatives of designs with given parameters t-(v,k,\lambda) and a given automorphism group A.

Firstly we present a method for computing the incidence matrices M^A_t,k by means of double cosets.

Solving the above system of equations is an NP-complete problem. We use a heuristic approach and represent the set of all solutions implicitly by a graph. This gives us the possibility either to extract the solutions explicitly, if there are not too many of them, or to compute their numbers.

Finally we can construct the isomorphism types of t-designs with given parameters and given automorphism group A, if we know about the structure of overgroups of A, or, if there are too many designs, we are in many cases still able to give the precise numbers.

With the help of the complete algorithm we verify many prominent results. To the best of our knowledge our approach for the first time allowed to compute the precise number of isomorphism types or even these designs themselves for substantial numbers; see the examples and tables at the end of this publication.

Lehrstuhl II für Mathematik, Universität Bayreuth

An extended version of this paper has been finally published under the title "The t-designs with prescribed automorphism group, new simple 6-designs" in J. Combin. Des. 1 (1993), 125-170.