The reversibility of (2r+1)-cyclic rule cellular automata

Abstract

In this paper, we introduce a family of one dimensional finite linear cellular automata with periodic boundary condition over primitive finite fields with p elements (Z,) which leads to a generalization in two directions: the radius and the field that states take values. This family of cellular automata is called (2r + 1)-cyclic cellular automata since it has a cyclic structure and its radius is r. Here, we establish a connection between the generator matrices of cyclic codes and the rule matrix of (2r + 1)-cyclic cellular automata. Thus this enables the determination of the reversibility problem of this cellular automaton by means of the algebraic coding theory. Further, we explicitly determine the reverse CA of this family and prove that the reverse CA of this family again falls into this family.