Convolutional codes over finite chain rings, MDP codes and their characterization

In this paper, we develop the theory of convolutional codes over finite commutative chain rings. In particular, we focus on maximum distance profile (MDP) convolutional codes and we provide a characterization of these codes, generalizing the one known for fields. Moreover, we relate MDP convolutional codes over a finite chain ring with MDP convolutional codes over its residue field. Finally, we provide a construction of MDP convolutional codes over finite chain rings generalizing the notion of superregular matrices.