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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.

Cyclic and dihedral constructions of even order

Aleš Drápal (2003)

Commentationes Mathematicae Universitatis Carolinae

Let G ( ∘ ) and G ( * ) be two groups of finite order n , and suppose that they share a normal subgroup S such that u ∘ v = u * v if u ∈ S or v ∈ S . Cases when G / S is cyclic or dihedral and when u ∘ v ≠ u * v for exactly n 2 / 4 pairs ( u , v ) ∈ G × G have been shown to be of crucial importance when studying pairs of 2-groups with the latter property. In such cases one can describe two general constructions how to get all possible G ( * ) from a given G = G ( ∘ ) . The constructions, denoted by G [ α , h ] and G [ β , γ , h ] , respectively, depend on a coset α (or two cosets β and γ ) modulo S , and on an...

Distinct equilateral triangle dissections of convex regions

Diane M. Donovan, James G. Lefevre, Thomas A. McCourt, Nicholas J. Cavenagh (2012)

Commentationes Mathematicae Universitatis Carolinae

We define a proper triangulation to be a dissection of an integer sided equilateral triangle into smaller, integer sided equilateral triangles such that no point is the vertex of more than three of the smaller triangles. In this paper we establish necessary and sufficient conditions for a proper triangulation of a convex region to exist. Moreover we establish precisely when at least two such equilateral triangle dissections exist. We also provide necessary and sufficient conditions for some convex...

Embedding 3 -homogeneous latin trades into abelian 2 -groups

Nicholas J. Cavenagh (2004)

Commentationes Mathematicae Universitatis Carolinae

Let T be a partial latin square and L be a latin square with T ⊆ L . We say that T is a latin trade if there exists a partial latin square T ' with T ' ∩ T = ∅ such that ( L ∖ T ) ∪ T ' is a latin square. A k -homogeneous latin trade is one which intersects each row, each column and each entry either 0 or k times. In this paper, we show the existence of 3 -homogeneous latin trades in abelian 2 -groups.

Equivalence classes of Latin squares and nets in ℂ P 2

Corey Dunn, Matthew Miller, Max Wakefield, Sebastian Zwicknagl (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

The fundamental combinatorial structure of a net in ℂ P 2 is its associated set of mutually orthogonal Latin squares. We define equivalence classes of sets of orthogonal Latin squares by label equivalences of the lines of the corresponding net in ℂ P 2 . Then we count these equivalence classes for small cases. Finally we prove that the realization spaces of these classes in ℂ P 2 are empty to show some non-existence results for 4-nets in ℂ P 2 .

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