Latin hypercubes with restricted transversals

Abstract

A $k$-plane of a $d$-dimensional array is a subarray formed by fixing $d-k$ coordinates and allowing the remaining $k$ coordinates to vary freely. A Latin hypercube of dimension $d$ and order $n$ is an $n\times n\times\cdots\times n$ array of dimension $d$ containing symbols from an $n$-set, such that each $1$-plane contains each of the possible entries exactly once. A transversal in a Latin hypercube of order $n$ is a set of $n$ entries of the hypercube, no pair of which agree in any coordinate or contain the same symbol. The aim of this paper is to construct Latin hypercubes that have transversals but which have many entries that are not in any transversal, or for which the number of disjoint transversals is limited. We show the following results in the case when the dimension $d$ is even. For all even $n\ge 10$ there exists a Latin hypercube of order $n$ that contains a transversal but for which all transversals hit one $(d-2)$-plane. For $n\in\{6,8\}$ there exists a Latin hypercube of order $n$ that contains a transversal but for which all transversals hit one of two $(d-2)$-planes. For even $d>2$ there is a Latin hypercube of order $n=4$ that contains a transversal but has $2^d$ entries that are not in any transversal. Our constructions use a quasigroup $(Q,\ast)$ to increase the dimension of a Latin hypercube using the rule $H_d(x_1,\dots,x_d)=H_{d-1}(x_1,\dots,x_{d-1})\ast x_d$. We give several characterisations which allow us to diagnose which entries of $H_d$ are in transversals in terms of properties of $H_{d-1}$ and $Q$.