The t‐designs with prescribed automorphism group, new simple 6‐designs

We introduce an algorithm for the construction of a complete system of representatives of t ‐designs with given parameters t − (v, k , λ) and prescribed full automorphism group A . It is based on the following observation published by Kramer and Mesner in 1976: The t − (v, k , λ) designs admitting automorphism group A are exactly the 0‐1‐solutions \documentclass{article}\pagestyle{empty}\begin{document}$ \overrightarrow x $\end{document} of the following system of linear equations\documentclass{article}\pagestyle{empty}\begin{document}$$ M_{t,k}^A \,\overrightarrow x = \left({\lambda \ldots \lambda} \right)^t. $$\end{document}M   t,k Aare incidence matrices, which we compute by means of double cosets. Representing the set of all solutions of the above system of equations implicitly by a graph gives us the possibility either to extract the solutions explicitly or to compute their precise numbers, which often are very big. We use the lattice of overgroups of A in the full symmetric group S v for the construction or enumeration of the isomorphism types of the t ‐designs with full automorphism group A from these solutions. To the best of our knowledge our approach for the first time allows one to compute the precise number of isomorphism types or even these designs themselves for substantial numbers. We determined the (number of) isomorphism types for many known parameter sets and found new simple 6‐designs with parameters\documentclass{article}\pagestyle{empty}\begin{document}$$ 6 - \left({28,8,\lambda} \right),\lambda = 42,63,84,105 $$\end{document}and full automorphism group PΓL 2 (27). We constructed all isomorphism types of these designs; their precise numbers are 3,367,21 743,38 277, respectively. © 1993 John Wiley & Sons, Inc.