# Mutually Orthogonal Latin Squares based on Cellular Automata

## TL;DR

This paper explores how Cellular Automata over finite fields can generate mutually orthogonal Latin squares, providing conditions for orthogonality, enumeration methods, and constructions for maximal sets of MOLS.

## Contribution

It establishes a link between Linear Bipermutive CA and orthogonal Latin squares, offering new enumeration and construction techniques for MOLS based on polynomial properties.

## Key findings

- Orthogonality of Latin squares corresponds to coprimality of local rule polynomials.
- Enumeration of MOLS pairs via counting coprime polynomials over finite fields.
- Construction of maximal MOLS sets using LBCA with irreducible polynomials.

## Abstract

We investigate sets of Mutually Orthogonal Latin Squares (MOLS) generated by Cellular Automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter $d$ over an alphabet of $q$ elements generates a Latin square of order $q^{d-1}$, we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field $\mathbb{F}_q$ are orthogonal if and only if the polynomials associated to their local rules are relatively prime. Next, we enumerate all such pairs of orthogonal Latin squares by counting the pairs of coprime monic polynomials with nonzero constant term and degree $n$ over $\mathbb{F}_q$. Finally, we present a construction of MOLS generated by LBCA with irreducible polynomials and prove the maximality of the resulting sets, as well as a lower bound which is asymptotically close to their actual number.