The Dynamics of Group Codes: Dual Abelian Group Codes and Systems
<|reference_start|>The Dynamics of Group Codes: Dual Abelian Group Codes and Systems: 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.<|reference_end|>
@article{forney2004the,
    title={The Dynamics of Group Codes: Dual Abelian Group Codes and Systems},
    author={G. David Forney Jr. and Mitchell D. Trott},
    journal={IEEE Trans. Inform. Theory, vol. 50, pp. 2935-2965, Dec. 2004.},
    year={2004},
    doi={10.1109/TIT.2004.838340},
    archivePrefix={arXiv},
    eprint={cs/0408038},
    primaryClass={cs.IT math.IT}
}