Isomorph-Free Exhaustive Generation of Designs with Prescribed Groups of Automorphisms

Abstract

We develop an algorithm framework for isomorph-free exhaustive generation of designs admitting a group of automorphisms from a prescribed collection of pairwise nonconjugate groups, where each prescribed group has a large index relative to its normalizer in the isomorphism-inducing group. We demonstrate the practicality of the framework by producing a complete classification of the Steiner triple systems of order $21$ admitting a nontrivial automorphism group. The number of such pairwise nonisomorphic designs is $62336617$, where $958$ of the designs are anti-Pasch. We also develop consistency checking methodology for gaining confidence in the correct operation of the algorithm implementation.