Enumerating extensions of mutually orthogonal Latin squares

TL;DR

This paper investigates the enumeration and extension of mutually orthogonal Latin squares (MOLS), providing bounds on how they can be extended to larger sets and applying these results to related combinatorial designs like Sudoku.

Contribution

It introduces bounds on extending k-MOLS to (k+1)-MOLS for growing k, generalizing to gerechte designs, and offers near-tight bounds for large k.

Findings

Bounds are tight for constant k.

Upper bounds on total number of k-MOLS for all k.

Results extend to gerechte designs including Sudoku.

Abstract

Two $n \times n$ Latin squares $L_1, L_2$ are said to be orthogonal if, for every ordered pair $(x,y)$ of symbols, there are coordinates $(i,j)$ such that $L_1(i,j) = x$ and $L_2(i,j) = y$. A $k$-MOLS is a sequence of $k$ pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed $k$, log-asymptotically tight bounds on the number of $k$-MOLS. To study the situation when $k$ grows with $n$, we bound the number of ways a $k$-MOLS can be extended to a $(k+1)$-MOLS. These bounds are again tight for constant $k$, and allow us to deduce upper bounds on the total number of $k$-MOLS for all $k$. These bounds are close to tight even for $k$ linear in $n$, and readily generalize to the broader class of gerechte designs, which include Sudoku squares.