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$2-(n^2, 2n, 2n-1)$ designs; incidence structure; 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 $AG_3(4, 2)$ (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 ${\mathcal D}({\mathcal A}, 2)$.
[1] Beth T., Jungnickel, D, Lenz H.: Designs Theory. : Bibliographisches Institut, Mannheim–Wien. 1985. MR 0779284
[2] Caggegi A.: Uniqueness of $AG_3(4, 2)$. Italian Journal of Pure and Applied Mathematics 15 (2004), 9–16. Zbl 1175.05028
[3] Hanani H.: Balanced incomplete block designs and related designs. Discrete Math. 11 (1975), 255–369. MR 0382030 | Zbl 0361.62067
[4] Hughes D. R., Piper F. C.: Projective Planes. : Springer-Verlag, Berlin–Heidelberg–New York. 1982, second printing. MR 0333959
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