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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 ) .

A family of 4-designs on 26 points

Dragan M. Acketa, Vojislav Mudrinski (1996)

Commentationes Mathematicae Universitatis Carolinae

Using the Kramer-Mesner method, 4 - ( 26 , 6 , λ ) designs with P S L ( 2 , 25 ) as a group of automorphisms and with λ in the set { 30 , 51 , 60 , 81 , 90 , 111 } are constructed. The search uses specific partitioning of columns of the orbit incidence matrix, related to so-called “quasi-designs”. Actions of groups P S L ( 2 , 25 ) , P G L ( 2 , 25 ) and twisted P G L ( 2 , 25 ) are being compared. It is shown that there exist 4 - ( 26 , 6 , λ ) designs with P G L ( 2 , 25 ) , respectively twisted P G L ( 2 , 25 ) as a group of automorphisms and with λ in the set { 51 , 60 , 81 , 90 , 111 } . With λ in the set { 60 , 81 } , there exist designs which possess all three considered groups...

A Method for Classification of Doubly Resolvable Designs and Its Application

Zhelezova, Stela (2011)

Serdica Journal of Computing

This article presents the principal results of the Ph.D. thesis Investigation and classification of doubly resolvable designs by Stela Zhelezova (Institute of Mathematics and Informatics, BAS), successfully defended at the Specialized Academic Council for Informatics and Mathematical Modeling on 22 February 2010.The resolvability of combinatorial designs is intensively investigated because of its applications. This research focuses on resolvable designs with an additional property - they have resolutions...

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