Computer Science – Information Theory
Scientific paper
2004-08-16
IEEE Trans. Inform. Theory, vol. 50, pp. 2935-2965, Dec. 2004.
Computer Science
Information Theory
30 pages, 11 figures. To appear in IEEE Trans. Inform. Theory, 2004
Scientific paper
10.1109/TIT.2004.838340
Fundamental results concerning the dynamics of abelian group codes (behaviors) and their duals are developed. Duals of sequence spaces over locally compact abelian groups may be defined via Pontryagin duality; dual group codes are orthogonal subgroups of dual sequence spaces. The dual of a complete code or system is finite, and the dual of a Laurent code or system is (anti-)Laurent. If C and C^\perp are dual codes, then the state spaces of C act as the character groups of the state spaces of C^\perp. The controllability properties of C are the observability properties of C^\perp. In particular, C is (strongly) controllable if and only if C^\perp is (strongly) observable, and the controller memory of C is the observer memory of C^\perp. The controller granules of C act as the character groups of the observer granules of C^\perp. Examples of minimal observer-form encoder and syndrome-former constructions are given. Finally, every observer granule of C is an "end-around" controller granule of C.
Forney David G. Jr.
Trott Mitchell D.
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