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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)
##### Keywords:
affine plane; parallel classes; block design
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##### References:
 [1] Beth T., Jungnickel, D, Lenz H.: Designs Theory. : Bibliographisches Institut, Mannheim-Wien. 1985. [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. · Zbl 0361.62067 [4] Hughes D. R., Piper F. C.: Projective Planes. : Springer-Verlag, Berlin-Heidelberg-New York. 1982, second printing. · Zbl 0484.51011
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