G. David Forney

G. David Forney, Heide Gluesing-Luerssen

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. A realization is called observable if there is a one-to-one correspondence between codewords and configurations, and controllable if it has independent constraints. A linear or group realization is observable if and only if its dual is controllable. 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 behavior partitions into disconnected subbehaviors, but this property does not hold for non-trellis realizations. On a general graph, the support of an unobservable configuration is a generalized cycle.

G. David Forney

This paper is mainly a semi-tutorial introduction to elementary algebraic topology and its applications to Ising-type models of statistical physics, using graphical models of linear and group codes. It contains new material on systematic (n,k) group codes and their information sets; normal realizations of homology and cohomology spaces; dual and hybrid models; and connections with system-theoretic concepts such as observability, controllability, and input/output realizations.

G. David Forney

The Conti-Boston factorization theorem (CBFT) for linear tail-biting trellis realizations is extended to group realizations with a new and simpler proof, based on a controller granule decomposition of the behavior and known controllability results for group realizations. Further controllability results are given; e.g., a trellis realization is controllable if and only if its top (controllability) granule is trivial.

G. David Forney

Given a controllable discrete-time linear system C, a shortest basis for C is a set of linearly independent generators for C with the least possible lengths. A basis B is a shortest basis if and only if it has the predictable span property (i.e., has the predictable delay and degree properties, and is non-catastrophic), or alternatively if and only if it has the subsystem basis property (for any interval J, the generators in B whose span is in J is a basis for the subsystem C_J). The dimensions of the minimal state spaces and minimal transition spaces of C are simply the numbers of generators in a shortest basis B that are active at any given state or symbol time, respectively. A minimal linear realization for C in controller canonical form follows directly from a shortest basis for C, and a minimal linear realization for C in observer canonical form follows directly from a shortest basis for the orthogonal system C^\perp. This approach seems conceptually simpler than that of classical minimal realization theory.