Constructing Objects up to Isomorphism, Simple 9-Designs with Small Parameters

Group actions are reviewed as a tool for classifying combinatorial objects up to isomorphism. The objective is a general theory for constructing representatives of isomorphism types. Homomorphisms of group actions allow to reduce problem sizes step by step. In particular, classifying by stabilizer type, i.e. the automorphism group of the objects, is generalized to using only sufficiently large subgroups of stabilizers. So, less knowledge of the full subgroup lattice of the classifying group is needed. For single steps in the homomorphism decomposition, isomorphism problems are transformed into double coset problems in groups. New lower bounds are given for the number of long double cosets such that corresponding bounds for the number of objects with trivial automorphism group can be derived. The theory is illustrated by an account of recent work on the construction of t-designs including new results. Based on a computer search by DISCRETA several simple 8-designs and the first simple 9-designs with small parameters are presented. The automorphism group is ASL(3, 3) acting on 27 and 28 points. There are many isomorphism types in each case. The number of isomorphism types is determined in the smaller cases. By relating the isomorphism types of design extensions to double cosets designs with small automorphism groups are also accessible. There result more than 1016 isomorphism types of 8-(28, 14, λ′) designs from each 8(27,13, λ) design. There are exactly 131,210,855,332,052,182,104 isomorphism types of 7-(25,9,45) designs obtained from extending all the 7-(24,8,5) designs with automorphism group PSL(2, 23) by all the 7-(24,9,40) designs with automorphism group PGL(2, 23). Most of these designs have a trivial automorphism group. Iterating forming extensions then results in more than 1062 isomorphism types of 7-(26,10,342) designs.