Observability, Controllability and Local Reducibility of Linear Codes on Graphs

TL;DR

This paper investigates the conditions under which linear code realizations on graphs are minimal, observable, controllable, and locally reducible, providing characterizations for various types of realizations including tail-biting trellises.

Contribution

It offers a comprehensive analysis of local reducibility, observability, and controllability of linear code realizations on graphs, including new characterizations for tail-biting trellises.

Findings

A realization is minimal iff all constraints are trim and proper on cycle-free graphs.

Unobservable or uncontrollable realizations are locally reducible.

Tail-biting trellis realizations are uncontrollable iff their trajectories form disconnected subrealizations.

Abstract

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.