Transitive Sets of Mutually Orthogonal Latin Squares

TL;DR

This paper explores the properties of transitive mutually orthogonal Latin squares (MOLS), confirming MacNeish's conjecture for simply transitive cases, and introduces new constructions and partial results for the broader transitive case.

Contribution

It provides the first confirmation of MacNeish's conjecture for simply transitive MOLS and introduces a novel construction method for MOLS through computational search.

Findings

MacNeish's conjecture holds for simply transitive MOLS.

No large transitive MOLS violating the conjecture were found.

A new computational construction method for MOLS was developed.

Abstract

We investigate MacNeish's conjecture (known to be false in general) in the setting of what we call "transitive" Mutually Orthogonal Latin Squares (MOLS). When we restrict our attention to "simply transitive" MOLS, we find that the conjecture holds. We provide some partial results towards the transitive case, as well as the outcome of a computer search, which introduces a new construction of MOLS. In particular, we were unable to find any transitive large (conjecture-violating) sets of MOLS in the literature.