Displaying similar documents to “On orthogonal Latin p -dimensional cubes”

Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

Diane M. Donovan, Mike Grannell, Emine Ş. Yazıcı (2020)

Commentationes Mathematicae Universitatis Carolinae

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We review results for the embedding of orthogonal partial Latin squares in orthogonal Latin squares, comparing and contrasting these with results for embedding partial Latin squares in Latin squares. We also present a new construction that uses the existence of a set of t mutually orthogonal Latin squares of order n to construct a set of 2 t mutually orthogonal Latin squares of order n t .

Overlapping latin subsquares and full products

Joshua M. Browning, Petr Vojtěchovský, Ian M. Wanless (2010)

Commentationes Mathematicae Universitatis Carolinae

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We derive necessary and sufficient conditions for there to exist a latin square of order n containing two subsquares of order a and b that intersect in a subsquare of order c . We also solve the case of two disjoint subsquares. We use these results to show that: (a) A latin square of order n cannot have more than n m n h / m h subsquares of order m , where h = ⌈ ( m + 1 ) / 2 ⌉ . Indeed, the number of subsquares of order m is bounded by a polynomial of degree at most 2 m + 2 in n . (b) For all n ≥ 5 there exists a loop of order...

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

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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 .