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

Andrea Caggegi (2006)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The simple incidence structure 𝒟 ( 𝒜 , 2 ) formed by points and unordered pairs of distinct parallel lines of a finite affine plane 𝒜 = ( 𝒫 , ) of order n > 2 is a 2 - ( n 2 , 2 n , 2 n - 1 ) design. If n = 3 , 𝒟 ( 𝒜 , 2 ) is the complementary design of 𝒜 . If n = 4 , 𝒟 ( 𝒜 , 2 ) is isomorphic to the geometric design A G 3 ( 4 , 2 ) (see [2; Theorem 1.2]). In this paper we give necessary and sufficient conditions for a 2 - ( n 2 , 2 n , 2 n - 1 ) design to be of the form 𝒟 ( 𝒜 , 2 ) for some finite affine plane 𝒜 of order n > 4 . As a consequence we obtain a characterization of small designs 𝒟 ( 𝒜 , 2 ) .

Biplanes.

Peter J. Cameron (1973)

Mathematische Zeitschrift

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