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arXiv · 2502.08868

Extendibility of Latin Hypercuboids

Candida Bowtell·Alice Devillers·André Kündgen·Padraig Ó Catháin·Ian M. Wanless
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Abstract

A Latin hypercuboid of order $n$ is a $d$-dimensional matrix of dimensions $n\times n\times\cdots\times n\times k$, with symbols from a set of cardinality $n$ such that each symbol occurs at most once in each axis-parallel line. If $k=n$ the hypercuboid is a Latin hypercube. The Latin hypercuboid is \emph{completable} if it is contained in a Latin hypercube of the same order and dimension. It is \emph{extendible} if it can have one extra layer added. In this note we consider which Latin hypercuboids are completable/extendible. We also consider a generalisation that involves multidimensional arrays of sets that satisfy certain balance properties. The extendibility problem corresponds to choosing representatives from the sets in a way that is analogous to a choice of a Hall system of distinct representatives, but in higher dimensions. The completability problem corresponds to partitioning the sets into such SDRs. We provide a construction for such an array of sets that does not have the property analogous to completability. A related concept was introduced by H\"aggkvist under the name $(m,m,m)$-array. We generalise a construction of $(m,m,m)$-arrays credited to Pebody, but show that it cannot be used to build the arrays that we need.

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Candida Bowtell, Alice Devillers, André Kündgen, Padraig Ó Catháin, Ian M. Wanless. 2025-02-13. Extendibility of Latin Hypercuboids. https://arxiv.org/abs/2502.08868

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