A 4-states algebraic solution to linear cellular automata synchronization

In this paper, we aim to present a completely new solution to the firing squad synchronization problem based on Wolfram's rule 60. This solution solves the problem on an infinite number of lines but not all possible lines. The two remarkable properties are that the state complexity of it is the lowest possible, say 4 states and 32 transitions (we prove that no line of length n⩾5 can be synchronized with only 3 states) and that the algorithm is no more based on geometric constructions but relies on some algebraic properties of the transition function. The solution is almost in minimal time: up to one unit of time.