Sparse Moore-local realisations of binary irreducible polynomials on near-square lattice regions
Year: 2026
Type: preprint
Abstract: For every $N\ge36$, we realise any prescribed monic irreducible binary polynomial of degree $N$ as the characteristic polynomial of a linear hybrid cellular automaton on a near-square region of exactly $N$ cells. The transition matrix is Moore-local with a null boundary and has at most $3N-1$ directed nonself dependencies. The dependency graph retains a bidirectional Hamilton path; its indegree, outdegree and underlying undirected degree are at most six. The underlying graph contains an explicit square grid minor of side proportional to $\sqrt N$. The deterministic synthesis takes $O(N^3)$ bit operations. The construction combines a local similarity transformation with a transport potential and joint routing across consecutive row gaps. Exact verification of finite certificates, followed by induction, proves the required routing inequalities for every admissible width. We also give an entirely analytic construction with fewer than $7N/2$ dependencies and prove a lower bound of $5N/2-O(\sqrt N)$ for the retained path and full rectangular grid minor. This lower bound is sharp when the characteristic polynomial is unrestricted.
Source: arXiv (Cornell University)
Author: Chen, Lizhong
Institution: Hong Kong University of Science and Technology
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220912226