Convolutional codes over groups

The basic algebraic structure theory of convolutional codes and their trellises is developed simultaneously for codes over groups, rings, and fields. The first part, which covers fundamental notions such as minimality and observability, is semi-tutorial in that most definitions are already standard (within the modern behavioral theory), as are some of the formally stated results. However, some of the pivotal results-emphasizing the role of observability as the basic well-behavedness condition for codes-are new, and several previous results are given simplified proofs. The usefulness of the behavioral approach even for convolutional codes over fields is demonstrated by a new minimality test for encoders as well as by the straightforward derivation of some known minimality criteria for generator matrices from the basic minimality criteria for group trellises. The second part of the paper deals with issues that are specific to codes over rings and groups. The main result is a concise characterization-the first such-of those groups that can appear as the branch group of any group trellis. It is further shown how such groups are “presented” by shift registers. A new large class of noncommutative convolutional codes is also given