Dynamical systems and convolutional codes over finite Abelian groups

Polynomial algebraic techniques have always played a central role in linear systems theory and also in the theory of convolutional codes. We show how such techniques can be generalized to study systems and codes defined over Abelian groups. The systems are considered from the “behavioral” point of view as developed by Willems in the 1980s, and some of our results can be seen as extensions of Willems' results to group systems. We also address a certain number of coding-oriented questions, and we propose concrete methods based on these algebraic techniques for the synthesis of encoders, inverters, and syndrome formers for codes over finite Abelian groups.