Rotational invariance of trellis codes. I. Encoders and precoders

We present a theoretical framework for rotational invariance of trellis codes. The distinction between codes and encoders plays a pivotal role. Necessary and sufficient conditions for rotational invariance are derived under general assumptions, and a construction is presented that obtains a rotationally invariant encoder for almost any rotationally invariant code, independent of the code´s algebraic structure. Encoders that use a differential precoder are considered as a separate case, where a system-theoretic characterization of precoding is used to find two alternative and slightly less general encoder constructions