A Method for Constructing Quasimultiple Affine Planes of Arbitrary Order

A quasimultiple affine plane of order n and multiplicity λ is a (v, k, λ) = (n2, n, λ) balanced incomplete block design. For those cases where the Bruck-Ryser theorem rules out λ = 1 it is an interesting problem to determine the smallest actual value of λ. Jungnickel [DiscreteMath. 85 (1990), 177–189] has conjectured that λ = 2 is possible for all orders. In this note we present a construction based on a result of Stahly which reduces a number of the bounds on λ given by Jungnickel. In particular, we construct quasidouble affine planes of orders 6 and 28 verifying the conjecture in these two cases.