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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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