### $2-({n}^{2},2n,2n-1)$ designs obtained from affine planes

The simple incidence structure $\mathcal{D}(\mathcal{A},2)$ formed by points and unordered pairs of distinct parallel lines of a finite affine plane $\mathcal{A}=(\mathcal{P},\mathcal{L})$ of order $n>2$ is a $2-({n}^{2},2n,2n-1)$ design. If $n=3$, $\mathcal{D}(\mathcal{A},2)$ is the complementary design of $\mathcal{A}$. If $n=4$, $\mathcal{D}(\mathcal{A},2)$ is isomorphic to the geometric design $A{G}_{3}(4,2)$ (see [2; Theorem 1.2]). In this paper we give necessary and sufficient conditions for a $2-({n}^{2},2n,2n-1)$ design to be of the form $\mathcal{D}(\mathcal{A},2)$ for some finite affine plane $\mathcal{A}$ of order $n>4$. As a consequence we obtain a characterization of small designs $\mathcal{D}(\mathcal{A},2)$.