On orthogonal Latin $p$-dimensional cubes

Marián Trenkler

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

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 $\frac{n}{m} (\binom{n}{h}) / (\binom{m}{h})$ 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 $\sqrt{2 m} + 2$ in $n$ . (b) For all $n \geq 5$ 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 $ℂ P^{2}$ .