Enumeration of MOLS of small order

TL;DR

This paper presents a comprehensive computer-based analysis of mutually orthogonal Latin squares (MOLS) for small orders, including enumeration, classification, and the discovery of near-MOLS for order 10.

Contribution

It provides the first detailed enumeration and classification of MOLS for orders up to 9, and reports the closest known set of MOLS for order 10.

Findings

Number of orthogonal mates for each Latin square of order n

Proportion of Latin squares with orthogonal mates for each n

A triple of Latin squares of order 10 close to being MOLS

Abstract

We report the results of a computer investigation of sets of mutually orthogonal latin squares (MOLS) of small order. For $n\le9$ we 1. Determine the number of orthogonal mates for each species of latin square of order $n$. 2. Calculate the proportion of latin squares of order $n$ that have an orthogonal mate, and the expected number of mates when a square is chosen uniformly at random. 3. Classify all sets of MOLS of order $n$ up to various different notions of equivalence. We also provide a triple of latin squares of order 10 that is the closest to being a set of MOLS so far found.