This paper explores cellular automata (CA) constructed from Yang–Baxter maps over finite fields <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {F}_{2^n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mi>n</mi> </msup> </msub> </math></EquationSource> </InlineEquation>. We define R-matrices using a map f on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {F}_{2^n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mi>n</mi> </msup> </msub> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {F}_{2^n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mi>n</mi> </msup> </msub> </math></EquationSource> </InlineEquation>, supported by analytical proofs for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. Our results further indicate that bijectivity is a fundamental requirement for this periodic behavior.

Integrable Cellular Automata on Finite Fields of Order $2^n$

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2026年6月22日

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Aoi Araoka,
Tetsuji Tokihiro

摘要

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

Integrable Cellular Automata on Finite Fields of Order \(2^n\)

摘要