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

2026-05-15

AI Summary

This study investigates whether Latin squares generated by nonlinear bipermutive cellular automata admit orthogonal mates, equivalently, whether they can be partitioned into disjoint transversals. Focusing on Latin squares of order \(N = 2^{d-1}\) induced by bipermutive rules of diameter \(d\) under null boundary conditions, the work establishes—for the first time without assuming linearity of the local rule—a necessary and sufficient condition for the main diagonal to form a transversal: this holds if and only if the associated cellular automaton with periodic boundary conditions is reversible over the configuration space of length \(d-1\). Combining algebraic characterizations with exhaustive search, the authors verify this criterion and demonstrate that \(d = 6\) is the smallest diameter admitting nonlinear solutions, thereby revealing the feasibility of constructing orthogonal Latin squares via nonlinear bipermutive cellular automata.

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.

Problem

• Latin squares
• transversals
• bipermutive cellular automata
• orthogonal mate
• nonlinear

Innovation

• bipermutive cellular automata
• Latin squares
• transversals
• orthogonal mates
• nonlinear local rules