On completion of latin hypercuboids of order 4

TL;DR

This paper proves that any latin hypercuboid of order 4 can be extended to a complete latin hypercube, advancing understanding of combinatorial array completion.

Contribution

It establishes the completion property for latin hypercuboids of order 4, a significant step in combinatorial design theory.

Findings

Any latin hypercuboid of order 4 is completable to a latin hypercube.

Provides a constructive proof for hypercuboid completion.

Enhances knowledge of array completion in combinatorics.

Abstract

A latin hypercuboid of order $N$ is an $N\times...\times N\times k$ array filled with symbols from the set $\{0,...,N-1\} $ in such a way that every symbol occurs at most once in every line. If $k=N$, such an array is a latin hypercube. We prove that any latin hypercuboid of order 4 is completable to a latin hypercube. Keywords: latin hypercube, n-ary quasigroup