Linear Cellular Automata and Fischer Automata

We study the sizes of minimal finite state machines associated with linear cellular automata. In particular, we construct a class of binary linear cellular automata whose corresponding minimal automata exhibit full exponential blow-up. These cellular automata have Hamming distance 1 to a permutation automaton. Moreover, the corresponding minimal Fischer automata as well as the minimal DFAs have maximal complexity. By contrast, the complexity of higher iterates of a cellular automaton always stays below the theoretical upper bound.