System-theoretic properties of convolutional codes over rings

Convolutional codes over rings are particularly suitable for representing codes over phase modulation signals. In order to develop a complete structural analysis of this class of codes, it is necessary to study rational matrices over rings, which constitutes the generator matrices (encoders) for such convolutional codes. Noncatastrophic, minimal, systematic and basic generator matrices are introduced and characterized by using a canonical form for polynomial matrices over rings. Finally some classes of convolutional codes, defined according to the generator matrix they admit, are introduced and analyzed from a system theoretic point of view. Index Terms Dynamical systems over rings, convolutional codes over rings, encoders, rational matrices, polynomial matrices.