Minimal syndrome formers for group codes

Given a controllable group code, it has been shown by Forney (1970, 1973), Trott, and Loeliger (1994) that it is possible to construct a canonical encoder whose state space coincides with the canonical state space of the code, that is the essential element determining the minimal trellis of the code. The construction of such an encoder is based on the concept of controllability granule. In this paper the construction of a syndrome former for an observable group code is proposed. This syndrome former exhibits analogous properties of the above mentioned canonical encoder and, in particular, its state space coincides with the canonical state space of the code. The proposed construction is based on the concept of observability granule, introduced by Forney and Trott (1993), which dualizes the concept of controllability granule. Similarly to what happens for the encoders, each observability granule produces a map and all these maps together provide the syndrome former. The main difference is that here, to achieve the right state-space dimension, the automata associated with these maps do not evolve independently from each other, but are coupled according to a triangular structure.