Computer Science – Information Theory
Scientific paper
2012-03-14
Computer Science
Information Theory
26 pages. Submitted to IEEE Transactions on Information Theory
Scientific paper
This paper investigates properties of realizations of linear or group codes on general graphs that lead to local reducibility. Trimness and properness are dual properties of constraint codes. A linear or group realization with a constraint code that is not both trim and proper is locally reducible. A linear or group realization on a finite cycle-free graph is minimal if and only if every local constraint code is trim and proper. The dual property to observability is the property of having independent constraints, which is called controllability. A simple counting test for controllability is given. An unobservable or uncontrollable realization is locally reducible. Parity-check realizations are controllable if and only if they have independent parity checks. In an uncontrollable tail-biting trellis realization, the trajectories partition into disconnected subrealizations, but this property does not hold for non-trellis realizations. On a general graph, the support of an unobservable trajectory is a generalized cycle.
Forney David G. Jr.
Gluesing-Luerssen Heide
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