Observability, controllability and local reducibility of linear codes on graphs

This paper is concerned with the local reducibility properties of linear realizations of codes on finite graphs. Trimness and properness are dual properties of constraint codes. A linear realization is locally reducible if any constraint code is not both trim and proper. On a finite cycle-free graph, a linear realization is minimal if and only if every constraint code is both trim and proper. A linear realization is called observable if it is one-to-one, and controllable if all constraints are independent. Observability and controllability are dual properties. An unobservable or uncontrollable realization is locally reducible. A parity-check realization is uncontrollable if and only if it has redundant parity checks. A tail-biting trellis realization is uncontrollable if and only if its trajectories partition into disconnected subrealizations. General graphical realizations do not share this property.