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The simple incidence structure $𝒟(𝒜,2)$ formed by points and unordered pairs of distinct parallel lines of a finite affine plane $𝒜=(𝒫,ℒ)$ of order $n>2$ is a $2-(n^2,2n,2n-1)$ design. If $n=3$, $𝒟(𝒜,2)$ is the complementary design of $𝒜$. If $n=4$, $𝒟(𝒜,2)$ 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 $𝒟(𝒜,2)$ for some finite affine plane $𝒜$ of order $n>4$. As a consequence we obtain a characterization of small designs $𝒟(𝒜,2)$.

Given a finite additive abelian group $G$ and an integer $k$, with $3\le k\le \left|G\right|$, denote by ${𝒟}_k\left(G\right)$ the simple incidence structure whose point-set is $G$ and whose blocks are the $k$-subsets $C=\{c_1,c_2,\cdots ,c_k\}$ of $G$ such that $c_1+c_2+\cdots +c_k=0$. It is known (see [Caggegi, A., Di Bartolo, A., Falcone, G.: Boolean 2-designs and the embedding of a 2-design in a group arxiv 0806.3433v2, (2008), 1–8.]) that ${𝒟}_k\left(G\right)$ is a 2-design, if $G$ is an elementary abelian $p$-group with $p$ a prime divisor of $k$. From [Caggegi, A., Falcone, G., Pavone, M.: On the additivity of block...