Affine Distance-transitive Groups of Dimension One

Let V be a one-dimensionalFq-space and G be a subgroup in AΓL(V) ∼ = AΓL(1, q) containing the translations and acting primitively on the set of vectors in V. Let G0be the stabilizer in G of the zero vector, so thatG0 ≤ ΓL(V) ∼ = ΓL(1, q) and G = V:G0 . Suppose that Γ is a graph structure on V on which G acts distance-transitively. Such graphs of diameter at most 2 are known from the literature. Here, we show that if Γ has diameter at least 3, then q = 64, G0 ∼ = Z9: Z3or Z9: Z6and Γ is the Hamming graph H(4, 3).