Codes on Graphs: Fundamentals
arXiv:1306.6264
Abstract
This paper develops a fundamental theory of realizations of linear and group codes on general graphs using elementary group theory, including basic group duality theory. Principal new and extended results include: normal realization duality; analysis of systems-theoretic properties of fragments of realizations and their connections; "minimal = trim and proper" theorem for cycle-free codes; results showing that all constraint codes except interface nodes may be assumed to be trim and proper, and that the interesting part of a cyclic realization is its "2-core;" notions of observability and controllability for fragments, and related tests; relations between state-trimness and controllability, and dual state-trimness and observability.
32 pages, 22 figures. To appear in IEEE Transactions on Information Theory. Part of this paper was presented at the 2012 Allerton Conference