3. Coding and Modulation for the Gaussian Channel in the Absence or in the Presence of Fluctuations

Looking for systems that combine coding and multilevel modulation whose Euclidean distance is close to that which results in the average
from random coding, we consider the combination of MDS codes over a large size alphabet a one-to-one mapping of the alphabet into a
symmetric constellation, e.g., phase modulation. Its performance in the presence of additive Gaussian noise can be predicted from that
of the random coding, provided that the signal to noise ratio is small enough. The results exhibit the sphere-hardening phenomenon
whether or not amplitude fluctuations are present. Weighted demodulator output and soft decoding should be affected in order to achieve this performance.
Such a decoding can be done in principle according to previous work by Fang and Battail. A prohibitive complexity can be avoided only at the expense of strict optimality.