Constellation design in four dimensions under average power constraint

In this paper we address the problem of constellation design in four dimensional space (4D) under average power constraint. We compare the performance of best lattice constellations with those optimized without any constraint on their structure. Even though the lattice based constellations provide the best minimum distance, usually they do not allow for a good binary labelling. Therefore, in more realistic system scenarios, up to a 2 dB gain can be obtained by properly optimizing the constellation and the corresponding binary labelling. We also investigate the performance of 4D constellations obtained by the Cartesian product of two 2D constellations.