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

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.