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Construction of Mendelsohn designs by using quasigroups of ( 2 , q ) -varieties

Lidija Goračinova-Ilieva, Smile Markovski (2016)

Commentationes Mathematicae Universitatis Carolinae

Let q be a positive integer. An algebra is said to have the property ( 2 , q ) if all of its subalgebras generated by two distinct elements have exactly q elements. A variety 𝒱 of algebras is a variety with the property ( 2 , q ) if every member of 𝒱 has the property ( 2 , q ) . Such varieties exist only in the case of q prime power. By taking the universes of the subalgebras of any finite algebra of a variety with the property ( 2 , q ) , 2 < q , blocks of Steiner system of type ( 2 , q ) are obtained. The stated correspondence between Steiner...

Construction, properties and applications of finite neofields

Anthony Donald Keedwell (2000)

Commentationes Mathematicae Universitatis Carolinae

We give a short account of the construction and properties of left neofields. Most useful in practice seem to be neofields based on the cyclic group and particularly those having an additional divisibility property, called D-neofields. We shall give examples of applications to the construction of orthogonal latin squares, to the design of tournaments balanced for residual effects and to cryptography.

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