Relationship between PXCAs and LFSRs

Clarify the relationship between permutation XOR cellular automata (PXCAs) and linear feedback shift registers (LFSRs), including whether the characteristic polynomial of a PXCA transition matrix provides a bridge between the two frameworks.

Background

The paper analyzes permutation XOR cellular automata consisting of Rule 90 combined with a partial-shift permutation connection. The authors characterize long binary periodic orbits through the characteristic polynomial of the resulting transition matrix over GF(2), specifically when the polynomial has a primitive factor.

The authors explicitly identify the relationship between PXCAs and LFSRs as not yet fully clarified and suggest that the characteristic polynomial may connect the two frameworks. This remains unresolved despite the paper’s algebraic characterization of long binary periodic orbits.