Integrable cellular automata on finite fields of order $2^n$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911456950222848 |
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| author | Araoka, Aoi Tokihiro, Tetsuji |
| author_facet | Araoka, Aoi Tokihiro, Tetsuji |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17148 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Integrable cellular automata on finite fields of order $2^n$ Araoka, Aoi Tokihiro, Tetsuji Exactly Solvable and Integrable Systems Cellular Automata and Lattice Gases 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. |
| title | Integrable cellular automata on finite fields of order $2^n$ |
| topic | Exactly Solvable and Integrable Systems Cellular Automata and Lattice Gases |
| url | https://arxiv.org/abs/2602.17148 |