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

Caggegi, Andrea

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 45 (2006) no. 1, pp. 31-34.

Résumé

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 ${\mathcal D}({\mathcal A}, 2)$.

Détail

Classification : 05B05, 05B25, 51E15
Keywords: $2-(n^2, 2n, 2n-1)$ designs; incidence structure; affine planes