Extendibility of Latin Hypercuboids

A Latin hypercuboid of order n is a d-dimensional matrix of dimensions n ×  n × ··· × n ×  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 completable if it is contained in a Latin hypercube of the same order and dimension. It is 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 Rela ed 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.