The Dynamics of Group Codes: Dual Abelian Group Codes and Systems

TL;DR

This paper explores the duality and dynamics of abelian group codes, revealing how properties like controllability and observability are interconnected through Pontryagin duality, with implications for system design.

Contribution

It develops a comprehensive framework linking the dynamics of abelian group codes with their duals, including new insights into controllability, observability, and granule structures.

Findings

Duals of sequence spaces are defined via Pontryagin duality.

Controllability of a code corresponds to observability of its dual.

Examples of minimal observer and syndrome-former constructions are provided.

Abstract

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.