The permutation elementary cellular automata (PECAs) are three-layer discrete-time dynamical systems. The input to hidden layer corresponds to an elementary cellular automaton and the hidden to output layer is one-to-one permutation connection. The dynamics is described by an autonomous difference equation of binary vectors. Depending on the permutation connections and rules, the PECAs generate a variety of binary periodic orbits. Especially, we have discovered direct stable binary periodic orbits characterized by long period, strong stability, and very fast transient phenomena. As the main result, we have clarified all permutation connections that cause 7-dimensional direct stable binary periodic orbits. This result is exact and novel. As a first step to engineering applications, we have implemented a hardware prototype based on field-programmable gate arrays (FPGAs) and have confirmed typical periodic orbits experimentally.