Modulation/Coding System for a Gaussian Channel

We consider a modulation/coding system for a memoryless Gaussian channel. As in some known systems, the inner stage uses phase modulation or amplitude/phase modulation signals, and the outer stage uses binary error-correcting codes. The system partitions the given signal set into subsets with mapping of subsets by Gray code. Various finite-dimensional modulation/coding systems have been constructed, showing substantial coding-rate gains with known systems. The asymptotic properties of the system are also examined.