Existence results for translation nets

Summary

We consider (s, r; μ)-nets admitting a group G of auto-morphisms acting regularly on the point set and fixing each parallel class; such nets will be called translation nets (due to Sprague for μ = 1). Our main interest is in deriving bounds on the maximum possible value of r (given s and μ) subject to certain restrictions on G (e.g. for abelian groups). Translation nets are equivalent to a generalization of the congruence partitions defined by André. We prove a decomposition theorem in the case of nilpotent groups and show that here the problem may be reduced to finding the maximum value of r for (pⁱ, r; p^j)-translation nets with elementary abelian translation group; this is related to partial t-spreads. Using a result of Schulz, we show that every translation affine design has the parameters of an affine space or desarguesian affine plane and has an elementary abelian translation group. A similar result holds for symmetric translation nets with a nilpotent translation group.

INTRODUCTION AND PRELIMINARY KNOWLEDGE

In this paper we consider a generalization of the well-known nets of Bruck [4], [5] where non-parallel blocks intersect μ times (μ not necessarily equal to 1); we are interested in such structures admitting a translation group G (i.e. G acts regularly on the point set and fixes each parallel class).