Multidimensional signal sets through the shell construction for parallel channels

The authors derive a procedure to send r bits on M parallel channels. A decomposition of the best constellation in Z^M+(1/2, . . ., 1/2) is given in terms of the cross-products of lower dimensional shells of points. The proposed scheme can be used with good known coset codes to provide an alternate method of coded modulation. The results indicate that one can get good shaping gains for low encoder complexity. The method is also generalized for channels with unequal gains. The authors also find a significant performance advantage at fixed shaping gain, in certain cases, with respect to the recent Voronoi constellations in terms of peak-to-average power and constellation expansion