Constructions of Pairs of Orthogonal Latin Cubes

TL;DR

This paper constructs new pairs of orthogonal Latin cubes for specific orders, expanding the known existence spectrum and linking these structures to MDS codes and orthogonal arrays.

Contribution

It introduces constructions of orthogonal Latin cubes for previously unknown orders, notably 84, and connects these to MDS codes and orthogonal arrays.

Findings

Constructed pairs of orthogonal Latin cubes for orders 16(18i-1)+4 and 16(18i+5)+4.

Achieved minimal new parameters for orthogonal arrays, notably OA_1(3,5,84).

Extended the known existence spectrum of orthogonal Latin cubes.

Abstract

A pair of orthogonal latin cubes of order $q$ is equivalent to an MDS code with distance $3$ or to an ${\rm OA}_1(3,5,q)$ orthogonal array. We construct pairs of orthogonal latin cubes for a sequence of previously unknown orders $q_i=16(18i-1)+4$ and $q'_i=16(18i+5)+4$. The minimal new obtained parameters of orthogonal arrays are ${\rm OA}_1(3,5,84)$. Keywords: latin square, latin cube, MOLS, MDS code, block design, Steiner system, orthogonal array