Improvements for lower bounds of mutually orthogonal Latin squares of sizes $54$, $96$ and $108$

TL;DR

This paper improves the known lower bounds for the number of mutually orthogonal Latin squares of sizes 54, 96, and 108 by constructing specific permutation codes and difference matrices, also correcting a previous error.

Contribution

It introduces new constructions for MOLS of sizes 54, 96, and 108, enhancing existing bounds and correcting prior inaccuracies.

Findings

At least 8 MOLS of order 54 are constructed.

At least 10 MOLS of order 96 are constructed.

A correction is made to previous work for order 45.

Abstract

We will show that there are at least 8, 10 and 9 mutually orthogonal Latin squares (MOLS) of orders $n=54$, $96$ and $108$. The cases $n=54$ and $96$ are obtained by constructing separable permutation codes consisting of $8 \times 54$ and $10 \times 96$ codeword respectively; in addition, these codes respectively have lengths $54$, $96$ and minimum distances $53$, $95$. Here we will follow exactly the procedure given in \cite{JS2019}. The case $n=108$ is obtained by constructing a $(108,10,1)$ difference matrix. Also, an error in \cite{ACD} for $n=45$ will be corrected.