Integrable cellular automata on finite fields of order 2ⁿ

2026/02/19 by Aoi Araoka, Tetsuji Tokihiro · 1 voice
Physics and Astronomy · #nlin.CG #nlin.SI

paper · pdf · doi:10.1007/s11040-026-09569-9

Abstract

This paper explores cellular automata (CA) constructed from Yang-Baxter maps over finite fields F_2ⁿ. We define R-matrices using a map f on F_2ⁿ 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_2ⁿ, supported by analytical proofs for n=2 and n=3. Our results further indicate that bijectivity is a fundamental requirement for this periodic behavior.

Citations

• Solitons with nested structure over finite fields
• Discrete Integrable Equations over Finite Fields
• Yang-Baxter Maps from the Discrete BKP Equation
• Yang-Baxter maps and symmetries of integrable equations on quad-graphs
• Algebro-geometric solution of the discrete KP equation over a finite field out of a hyperelliptic curve
• The discrete KP and KdV equations over finite fields
• Yang-Baxter maps and integrable dynamics

Discussions

• Aoi Araoka, Tetsuji Tokihiro: Integrable cellular automata on finite fields of order $2^n$ https://arxiv.org/abs/2602.17148 https://arxiv.org/pdf/2602.17148 https://arxiv.org/html/2602.17148 [bsky, 0 points, 0 comments]

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