Extendibility of Latin Hypercuboids

TL;DR

This work surveys extendibility and completability of Latin hypercuboids, defines the key objects $LHC_d(n,k)$, $NE_d(n)$ and $NC_d(n)$, and frames the problems through SDR/Hall-type perspectives in higher dimensions. It generalizes to $(n^d,k)$-arrays, presents constructions yielding nonlayerable arrays, and situates these findings with respect to Häggkvist’s $(m,m,m)$-arrays and Pebody’s approaches. The paper collects the best-known bounds in dimension $d=3$, including $\mathrm{NC}_3(n)\le \lceil n/2\rceil$ and $\mathrm{NE}_3(n)\le (n-1)/2$ (when $n\equiv3\bmod4$) or $\le 3n/4+O(1)$ otherwise, while reporting newer linear bounds and density-based extendibility results in higher dimensions. It ends with open questions on growth rates, random models, and the impact of realisability, outlining several promising directions for future research in combinatorial design and higher-dimensional SDR theory.

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 Häggkvist 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.

Theorems & Definitions (13)

• theorem 1
• proof
• theorem 2
• theorem 3
• theorem 4
• theorem 5
• theorem 6
• theorem 7
• theorem 8
• proof