2-D Reversible Cellular Automata with Nearest and Prolonged Next Nearest Neighborhoods under Periodic Boundary

Abstract

In this paper, we study a 2-dimensional cellular automaton generated by a new local rule with the nearest neighborhoods and prolonged next nearest neighborhoods under periodic boundary condition over the ternary field (Z(3)). We obtain the rule matrix of this cellular automaton and characterize this family by exploring some of their important characteristics. We get some recurrence equations which simplifies the computation of the rank of the rule matrix related to the 2-dimensional cellular automaton drastically. Next, we propose an algorithm to determine the rank of the rule matrix. Finally, we conclude by presenting an application to error correcting codes.