On behavior of two-dimensional cellular automata with an exceptional rule

Several studies [A.R. Khan, P.P. Choudhury, K. Dihidar, S. Mitra, P. Sarkar, VLSI architecture of a cellular automata, Comput. Math. Appl. 33 (1997) 79–94; A.R. Khan, P.P. Choudhury, K. Dihidar, R. Verma, Text compression using two-dimensional cellular automata, Comput. Math. Appl. 37 (1999) 115–127; K. Dihidar, P.P. Choudhury, Matrix algebraic formulae concerning some exceptional rules of two-dimensional cellular automata, Inf. Sci. 165 (2004) 91–101] have explored a new rule convention for two-dimensional (2-D) nearest neighborhood linear cellular automata (CA) with null and periodic boundary conditions. A variety of applications of the rule convention have been illustrated, and the VLSI architecture of cellular automata machine (CAM) has been proposed. However, most of the studies address the issue of the kernel dimension of 2-D CA, and many other important characteristics of CA, such as Garden of Eden (GOE), maximal transient length, maximal cycle length, etc., have not been explored. In this paper, by exploiting matrix algebra in GF(2) (the Galois field with two elements), we attempt to characterize the behavior of a specific rule, which is not covered in existing work but accords with the same convention. A necessary and sufficient condition is given to ensure that a given configuration is a GOE. Meanwhile, we propose some algorithms to determine the number of GOEs, the maximal transient length, and the maximal cycle length in a 2-D CA.