On the transversals of Latin squares generated by nonlinear bipermutive cellular automata

TL;DR

This paper investigates conditions under which nonlinear bipermutive cellular automata generate Latin squares with transversals, focusing on the main diagonal and providing characterizations and experimental results.

Contribution

It characterizes when the main diagonal of Latin squares from nonlinear bipermutive CA forms a transversal and identifies the smallest diameter for such CA through experiments.

Findings

Main diagonal forms a transversal iff the generating function induces an invertible CA.

D=6 is the smallest diameter for nonlinear bipermutive CA with Latin squares having a main diagonal transversal.

Exhaustive search confirms the existence of such CA at D=6.

Abstract

In this short paper, we begin to investigate the conditions under which a generic Bipermutive Cellular Automaton (BCA) with no-boundary conditions of diameter $d$ generates a Latin square of order $N=2^{d-1}$ admitting an orthogonal mate, without relying on the linearity of the local rule. Since an orthogonal mate exists if and only if the Latin square can be partitioned into $N$ disjoint \emph{transversals}, we start by characterizing the subclass of BCA whose Latin squares have a transversal on the main diagonal. In particular, we prove that the main diagonal forms a transversal if and only if the generating function of the bipermutive local rule induces an invertible CA with periodic boundary conditions on a configuration of size $d-1$. We then perform exhaustive search experiments, showing that $d=6$ is the smallest diameter for which there exist nonlinear bipermutive CA that generate Latin squares with a transversal on the main diagonal.