Integrable cellular automata on finite fields of order $2^n$

TL;DR

This paper investigates cellular automata derived from Yang-Baxter maps over finite fields of order 2^n, establishing conditions for their construction and analyzing their periodic behavior.

Contribution

It provides necessary and sufficient conditions for Yang-Baxter maps over F_{2^n} and explores the periodicity of associated cellular automata, supported by exhaustive computational searches.

Findings

Streamlined conditions for Yang-Baxter maps in characteristic two

Large-scale enumeration of bijective solutions in small fields

Evidence linking bijectivity to periodic behavior in cellular automata

Abstract

This paper explores cellular automata (CA) constructed from Yang-Baxter maps over finite fields $F_{2^n}$. We define $R$-matrices using a map $f$ on $F_{2^n}$ and establish necessary and sufficient conditions for $f$ to satisfy the Yang-Baxter equation. We show that these conditions become remarkably streamlined in characteristic two. An exhaustive search for bijective solutions in fields of order 4, 8, and 16 yields 16, 736, and 269,056 maps, respectively. Analysis of the resulting CA under helical boundary conditions reveals a consistent alignment between the temporal period and the field order. We propose the conjecture that this periodic identity holds generally for $F_{2^n}$, supported by analytical proofs for $n=2$ and $n=3$. Our results further indicate that bijectivity is a fundamental requirement for this periodic behavior.