Pairs of MOLS of order ten satisfying non-trivial relations

TL;DR

This paper investigates pairs of mutually orthogonal Latin squares of order ten, revealing the extensive variety of non-trivial linear relations they satisfy and demonstrating that these pairs do not extend to larger sets.

Contribution

It provides a comprehensive enumeration of pairs of MOLS of order ten with non-trivial relations and shows these pairs cannot be extended to triples, advancing understanding of their structure.

Findings

Found 18,526,320 pairs of MOLS satisfying at least one non-trivial relation.

Proved that none of these pairs extend to a triple of MOLS.

Ruled out an additional relation on a set of 3-MOLS from previous classifications.

Abstract

A relation on a $k$-net$(n)$ (or, equivalently, a set of $k-2$ mutually orthogonal Latin squares of order $n$) is an $\mathbb{F}_{2}$ linear dependence within the incidence matrix of the net. Dukes and Howard (2014) showed that any 6-net(10) satisfies at least two non-trivial relations, and classified the relations that could appear in such a net. We find that, up to equivalence, there are $18\,526\,320$ pairs of MOLS satisfying at least one non-trivial relation. None of these pairs extend to a triple. We also rule out one other relation on a set of $3$-MOLS from Dukes and Howard's classification.