Symmetry properties of multilevel coded modulation

We consider the geometrical uniformity properties of signal space codes generated by multilevel coded modulation. First we examine multilevel schemes based on 2^L-ary PSK signal sets, and we derive conditions on the binary component codes which lead to geometrically uniform signal space codes. Next we study the geometrical uniformity of multilevel schemes based on signal sets having a generating group isomorphic to a direct product. Finally, we provide an easy-to-test criterion for finding if a given signal set (or signal space code) obtained as the union of two orbits of the same generating group has the uniform distance property