On behavior of two-dimensional cellular automata with an exceptional rule under periodic boundary condition

This article deals with the behavior of two-dimensional (2-D) cellular automata (CA) with a special rule under periodic boundary condition by using matrix algebra. The important characteristics of CA have been studied, such as Garden of Eden (GOE), maximal transient length, maximal cycle length and so forth. Several necessary and sufficient conditions are provided, which guarantee a given configuration of being a GOE in different cases. Besides, algorithms are proposed to obtain the number of GOEs, the maximal transient length, and the maximal cycle length in such a CA with the rule mentioned above under periodic condition.