Block-transitive designs in affine spaces

This paper deals with block-transitive t-(v, k, λ) designs in affine spaces for large t, with a focus on the important index λ = 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.