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.
convolutional codes over rings rational matrices encoders polynomial matrices dynamical systems over rings