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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.

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