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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 geometric approach to universal quasigroup identities

Václav J. Havel (1993)

Archivum Mathematicum

In the present paper we construct the accompanying identity I ^ of a given quasigroup identity I . After that we deduce the main result: I is isotopically invariant (i.e., for every guasigroup Q it holds that if I is satisfied in Q then I is satisfied in every quasigroup isotopic to Q ) if and only if it is equivalent to I ^ (i.e., for every quasigroup Q it holds that in Q either I , I ^ are both satisfied or both not).

A parallelogram configuration condition in nets

Jitka Markvartová (1993)

Archivum Mathematicum

After describing a (general and special) coordinatization of k -nets there are found algebraic equivalents for the validity of certain quadrangle configuration conditions in k -nets with small degree k .

Biembeddings of symmetric configurations and 3-homogeneous Latin trades

Mike J. Grannell, Terry S. Griggs, Martin Knor (2008)

Commentationes Mathematicae Universitatis Carolinae

Using results of Altshuler and Negami, we present a classification of biembeddings of symmetric configurations of triples in the torus or Klein bottle. We also give an alternative proof of the structure of 3-homogeneous Latin trades.

Binary codes and partial permutation decoding sets from the odd graphs

Washiela Fish, Roland Fray, Eric Mwambene (2014)

Open Mathematics

For k ≥ 1, the odd graph denoted by O(k), is the graph with the vertex-set Ωk, the set of all k-subsets of Ω = 1, 2, …, 2k +1, and any two of its vertices u and v constitute an edge [u, v] if and only if u ∩ v = /0. In this paper the binary code generated by the adjacency matrix of O(k) is studied. The automorphism group of the code is determined, and by identifying a suitable information set, a 2-PD-set of the order of k 4 is determined. Lastly, the relationship between the dual code from O(k)...

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