On 2 – (n2, 2n, 2n–1) designs with three intersection numbers

The simple incidence structure $${\mathcal{D}(\mathcal{A},2)}$$ , formed by the points and the unordered pairs of distinct parallel lines of a finite affine plane $${\mathcal{A}=(\mathcal{P}, \mathcal{L})}$$ of order n > 4, is a 2 – (n2,2n,2n–1) design with intersection numbers 0,4,n. In this paper, we show that the converse is true, when n ≥ 5 is an odd integer.