Decompositions of Kramer-Mesner matrices
Reinhard Laue · Contemporary mathematics - American Mathematical Society · 2014
Selecting orbits of a group G G on k k -sets of a point set V V that together form a t t - ( v , k , λ ) (v,k,\lambda ) design has been formalized by Kramer and Mesner in 1976. Since then, solving systems of Diophantine equations following their approach has become the standard method, resulting in t t -designs on small point sets up to t = 8 t = 8 . Most of the known Steiner 5 5 -designs, i.e. designs with λ = 1 \lambda = 1 , were found this way. The prescribed automorphism group was usually chosen as a highly transitive group. In this paper, we look at groups acting on structured sets, which need not be transitive. Instead we make use of the underlying structure to decompose the Kramer-Mesner matrices into submatrices. This is possible if the group G G has fixed points or if the point set can be interpreted as a cartesian product that is invariant under G G . In the latter case, there result new Diophantine systems of linear equations that can be addded to the Kramer-Mesner system. These can also be used to find constraints for possible subsystems of a desired design. Structuring the point set in this way allows to apply homomorphisms of group actions. These allow a local solution of subproblems. In particular, prescribing block stabilizers as homomorphic images can be used to restrict the search for t t -designs to only parts of a large Kramer-Mesner matrix. There result designs with fairly large parameters. A backtrack search is proposed that combines these approaches.