Algebraic and Computational Analysis of Elementary Cellular Automaton Rule 150

Elementary cellular automata provide simple mathematical models capable of generating complex spatiotemporal behavior. Rule 150 is an additive binary cellular automaton whose local update function is the exclusive OR (XOR) of a cell and its two nearest neighbors. Despite its simple definition, its finite-size dynamics can exhibit nontrivial periodic behavior. This research note presents an algebraic and computational investigation of Rule 150 on a one-dimensional periodic lattice. The system is represented as a linear transformation over the finite field F2, allowing its evolution to be studied using binary matrices and polynomial arithmetic. Computational experiments examined XOR conservation, finite-state recurrence, spatial propagation, population density, and the rank of the global update operator. The reported experiments showed that the XOR parity of the global state remained invariant in the tested simulations. For odd lattice sizes, rank calculations exhibited a consistent pattern: the global update matrix had full rank when the lattice size was not divisible by three, while its rank was two less than the lattice size when the size was divisible by three. This pattern admits an algebraic explanation through the polynomial representation of the update operator. Several observed recurrence times, including 3, 7, 31, 511, and 1023, were also recorded. These values must be distinguished from a general formula for the period, which has not been established here. Related period sequences and algebraic properties of cellular automata have been studied previously. The principal contribution of this note is to consolidate the computational observations and present a finite-field framework for analyzing Rule 150. The results establish a basis for further work on exact periods, operator orders, and the relationship between lattice size and dynamical recurrence.