Block-Transitive Designs in Affine Spaces

TL;DR

This paper proves the non-existence of certain highly symmetric block designs in affine spaces for large parameters, specifically for 4- and 5-$(v,k,1)$ designs with affine automorphism groups.

Contribution

It establishes new non-existence results for 4- and 5-$(v,k,1)$ designs with affine automorphism groups, advancing understanding of symmetry constraints in affine block designs.

Findings

No non-trivial 5-$(v,k,1)$ designs with affine automorphisms exist.

Non-existence of 4-$(v,k,1)$ designs holds except possibly for one-dimensional affine groups.

Analysis of finite 2-homogeneous affine permutation groups underpins the proofs.

Abstract

This paper deals with block-transitive $t$-$(v,k,\lambda)$ designs in affine spaces for large $t$, with a focus on the important index $\lambda=1$ case. We prove that there are no non-trivial 5-$(v,k,1)$ designs admitting a block-transitive group of automorphisms that is of affine type. Moreover, we show that the corresponding non-existence result holds for 4-$(v,k,1)$ designs, except possibly when the group is one-dimensional affine. Our approach involves a consideration of the finite 2-homogeneous affine permutation groups.