Characteristic and Minimal Polynomials of Linear Cellular Automata

The 1984 article by Martin, Odlyzko and Wolfram on Wolfram’s Rule 90 heightened interest in the area of linear cellular automata defined on a vector space over a finite field [8]. In this article we find the closed form expression for the minimal polynomial of Rule 90, which is then used to completely characterize the dynamics of the map. In addition, we show that the standard basis vectors in ZZ 2 lead to the maximal cycle for any linear finite dimensional cellular automata with periodic boundary conditions. Finally, we address questions posed by Tadaki on the connection between Rule 90 and Rule 150.