Mutually orthogonal latin squares with large holes

Peter J. Dukes; Christopher M. van Bommel

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arXiv · 1410.6743

Mutually orthogonal latin squares with large holes

Peter J. Dukes·Christopher M. van Bommel

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Abstract

Two latin squares are orthogonal if, when they are superimposed, every ordered pair of symbols appears exactly once. This definition extends naturally to `incomplete' latin squares each having a hole on the same rows, columns, and symbols. If an incomplete latin square of order $n$ has a hole of order $m$, then it is an easy observation that $n \ge 2m$. More generally, if a set of $t$ incomplete mutually orthogonal latin squares of order $n$ have a common hole of order $m$, then $n \ge (t+1)m$. In this article, we prove such sets of incomplete squares exist for all $n,m \gg 0$ satisfying $n \ge 8(t+1)^2 m$.

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Peter J. Dukes, Christopher M. van Bommel. 2014-10-24. Mutually orthogonal latin squares with large holes. https://arxiv.org/abs/1410.6743

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