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

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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

### Some Additive $2-\left(v,5,\lambda \right)$ Designs

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Given a finite additive abelian group $G$ and an integer $k$, with $3\le k\le |G|$, denote by ${𝒟}_{k}\left(G\right)$ the simple incidence structure whose point-set is $G$ and whose blocks are the $k$-subsets $C=\left\{{c}_{1},{c}_{2},\cdots ,{c}_{k}\right\}$ 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...

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