Permutation XOR cellular automata: partial-shift connections and half-primitive polynomials
Year: 2026
Type: preprint
Abstract: This paper presents a permutation XOR cellular automaton, a simple digital dynamical system composed of an elementary cellular automaton of the XOR rule and a partial-shift permutation connection characterized by a shift-part parameter. The proposed system is suitable for precise analysis and simple hardware implementation. Depending on the partial-shift connection and system dimension, the automaton can generate long binary periodic orbits (LBPOs) with strong stability. As a principal result, we identify the partial-shift parameters and dimensions that enable the generation of LBPOs. In our analysis, these LBPOs are characterize using half-primitive polynomials over $\mathrm{GF}(2)$. Furthermore, we present a simple FPGA-based hardware implementation. The hardware transforms typical LBPOs into electrical signals for potential engineering applications.
Source: arXiv (Cornell University)
Institution: Hosei University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214754338