Construction of minimal syndrome formers for group codes

System theoretic concepts have played in the past an important role in the development and in the analysis of error correcting codes. Indeed the concepts of encoder, syndrome former and of trellis for a code are strictly related with the notions of image representation, kernel representation and state space representation of a behavior, respectively. In previous years, the need of a joint optimization of the modulation and coding procedures produced the development of coding techniques (trellis coded modulation schemes) that can be often well modeled by resorting to the concept of group code, that is a code defined over a group alphabet. For this class of codes the existence of a canonical minimal trellis and of a canonical minimal encoder has been proved in Forney and Trott (1993). In this paper we complete the analysis provided in the above article showing that also a canonical minimal syndrome former exists and that it can always be constructed starting from the observability granules of the group code. These sets dualize the so called controllability granules which play an essential role in the construction of the canonical encoder proposed in the above article