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Observability, controllability and local reducibility of linear codes on graphs

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2 Author(s)
G. David Forney ; Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139 ; Heide Gluesing-Luerssen

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.

Published in:

Information Theory Proceedings (ISIT), 2012 IEEE International Symposium on

Date of Conference:

1-6 July 2012