Exhaustive Generation of Linear Orthogonal Cellular Automata

TL;DR

This paper develops an enumeration algorithm for all pairs of linear cellular automata over binary fields that produce orthogonal Latin squares, connecting algebraic, combinatorial, and language theory insights.

Contribution

It introduces a practical enumeration method for coprime polynomial pairs over GF(2), addressing a gap in previous theoretical counting results.

Findings

Enumeration algorithm for coprime polynomial pairs over GF(2)

Connections established with algebraic language theory and combinatorics

Alternative derivation of the counting formula for orthogonal CA pairs

Abstract

We consider the problem of exhaustively visiting all pairs of linear cellular automata which give rise to orthogonal Latin squares, i.e., linear Orthogonal Cellular Automata (OCA). The problem is equivalent to enumerating all pairs of coprime polynomials over a finite field having the same degree and a nonzero constant term. While previous research showed how to count all such pairs for a given degree and order of the finite field, no practical enumeration algorithms have been proposed so far. Here, we start closing this gap by addressing the case of polynomials defined over the field $\F_2$, which corresponds to binary CA. In particular, we exploit Benjamin and Bennett's bijection between coprime and non-coprime pairs of polynomials, which enables us to organize our study along three subproblems, namely the enumeration and count of: (1) sequences of constant terms, (2) sequences of degrees, and (3) sequences of intermediate terms. In the course of this investigation, we unveil interesting connections with algebraic language theory and combinatorics, obtaining an enumeration algorithm and an alternative derivation of the counting formula for this problem.