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\(2\)-\((n^2,2n,2n-1)\) designs obtained from affine planes. (English) Zbl 1125.05015
Consider the incidence structure \({\mathcal{D}}({\mathcal A}, 2)\) formed by points and unordered pairs of parallel lines of an affine plane \({\mathcal A}\) of order \(n > 2\) as blocks. This is easily seen to be a 2-\((n^2, 2n, 2n-1)\)-design. For \(n > 4\), the author gives an interesting characterization of designs with these parameters that necessarily arise out of an affine plane of order \(n\) in terms of the following two conditions: (c1) any three distinct points of \({\mathcal D}\) are contained in either three or \(n-1\) common blocks, and (c2) if \(X_1, X_2, \dots, X_{n-1}\) are distinct blocks of \({\mathcal D}\) such that \(| X_1 \cap X_2 \cap \dots \cap X_{n-1}| > 2\) then \(X_1 \cap X_2 \cap \dots \cap X_{n-1} = X_i \cap X_j\) for all \(i \neq j\). Proof uses two way counting of flags.

05B05 Combinatorial aspects of block designs
05B25 Combinatorial aspects of finite geometries
51E15 Finite affine and projective planes (geometric aspects)
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