Minimal and systematic convolutional codes over finite Abelian groups

Convolutional codes over Abelian groups provide an effective theoretical framework for the analysis of some classes of TCM codes. The encoder synthesis for this class of codes is not as simple as in the binary case, since minimal encoders in the group case might be necessarily nonlinear. In this contribution an algorithmic method testing whether a convolutional code over an Abelian group admits a systematic or a minimal homomorphic encoder is provided. This test consists in verifying whether a subgroup splits in a group. Through this method, the class of codes admitting systematic encoders and the class of codes admitting minimal encoders can be compared. Finally this test is applied to some examples of practical relevance.