Minimality tests for rational encoders over rings

Given an encoding matrix over some field, various criteria are known to check minimality. Most of these criteria apply to encoders of a particular class, e.g. basic encoders or systematic encoders, and only a few criteria are general in the sense that they apply to arbitrary rational encoding matrices. In this paper, causal rational encoders over commutative rings are considered and a general criterion of Johannesson and Wan (see IEEE Trans. Inform. Theory, vol.39, p.1219-33, July 1993) is generalized to rings, which satisfy the descending chain condition. Moreover, a new simple test is presented that reduces the minimality question from the ring to the field case. The basis for these new minimality tests are the concept of minimality of group systems and convolutional codes