Maximal Length Cellular Automata : A Survey

TL;DR

This survey reviews the theoretical foundations of maximal length cellular automata, highlighting their connection to primitive polynomials, and provides new insights, results, and open problems for researchers.

Contribution

It offers a comprehensive tutorial on maximal length CA theory, including new theorems and corollaries, bridging gaps among existing results.

Findings

Explores the link between maximal length CA and primitive polynomials.

Provides new theorems and corollaries related to maximal length CA.

Summarizes known results with references and open problems.

Abstract

This article surveys some theoretical aspects of Cellular Automata (CAs) research. In particular, we discuss on maximal length CA. An n-cell CA is a maximal length CA, if all the configurations except one form a single cycle. There is a bonding between maximal length CA and primitive polynomial. So, primitive polynomials occupy a good amount of space in this survey. The main goal of this survey is to provide a tutorial on maximal length CA theory to researchers with classical and new results on maximality. We also give a compact collection of known results with references to their proofs, and to suggest some open problems. Additionally, some new theorems and corollaries are added to bridge the gaps among several known results.