# Isometry invariant permutation codes and mutually orthogonal Latin squares

## TL;DR

This paper constructs new sets of mutually orthogonal Latin squares (MOLS) using isometry invariant permutation codes, providing improved lower bounds for the number of MOLS for specific orders and offering a computational approach for their description.

## Contribution

It introduces a novel method to construct MOLS via isometry invariant permutation codes, resulting in new lower bounds for certain orders and a uniform computational description.

## Key findings

- Constructed permutation codes of lengths 35, 48, 63, 96 with specified minimum distances.
- Derived new lower bounds for the number of MOLS for n=35, 48, 63, 96.
- Codes are described using subgroup generators of the isometry group, facilitating computational analysis.

## Abstract

Commonly the direct construction and the description of mutually orthogonal Latin squares (MOLS) makes use of difference or quasi-difference matrices. Now there exists a correspondence between MOLS and separable permutation codes. We like to present separable permutation codes of length $35$, $48$, $63$ and $96$ and minimum distance $34$, $47$, $62$ and $95$ consisting of $6 \times 35$, $10 \times 48$, $8 \times 63$ and $8 \times 96$ codewords respectively. Using the correspondence this gives $6$ MOLS for $n=35$, $10$ MOLS for $n=48$, $8$ MOLS for $n=63$ and $8$ MOLS for $n=96$. So $N(35) \ge 6$, $N(48) \ge 10$, $N(63) \ge 8$ and $N(96) \ge 8$ holds which are new lower bounds for MOLS. The codes will be given by generators of an appropriate subgroup $U$ of the isometry group of the symmetric group $S_n$ and $U$-orbit representatives. This gives an alternative uniform way to describe the MOLS where the data for the codes can be used as input for computer algebra systems like MAGMA, GAP etc.