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Some Additive 2 - ( v , 5 , λ ) Designs

Andrea Caggegi — 2015

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Given a finite additive abelian group G and an integer k , with 3 k | G | , denote by 𝒟 k ( G ) the simple incidence structure whose point-set is G and whose blocks are the k -subsets C = { c 1 , c 2 , , c k } of G such that c 1 + c 2 + + c k = 0 . It is known (see [Caggegi, A., Di Bartolo, A., Falcone, G.: Boolean 2-designs and the embedding of a 2-design in a group arxiv 0806.3433v2, (2008), 1–8.]) that 𝒟 k ( G ) is a 2-design, if G is an elementary abelian p -group with p a prime divisor of k . From [Caggegi, A., Falcone, G., Pavone, M.: On the additivity of block...

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 ) .

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