Journal of Statistical Physics, Vol. 43, Nos. 3/4, 1986. p. 463-478.
Deterministic cellular automata (CA) with additive rules are studied by exploiting the properties of circulant matrices on finite fields. Complete state transition diagrams for higher-order and multidimensional CA on finite lattices are analyzed. Conditions on the rules which make them reversible are obtained. It is shown that all state transition diagrams of the CA have identical trees rooted on cycles. General formulae for cycle lengths and multiplicities are given.