A General Framework for Geometrically Uniform Codes and Signal Sets Matched to Groups

Geometrically uniform codes are fundamental in communication systems, mainly for modulation. Typically, geometrically uniform codes are dependent on a given alphabet. The current work establishes the necessary and sufficient conditions for obtaining a matched labeling between a group G and a signal set S. It introduces the concept of the G-isometric signal set, allowing for the establishment of equivalences between different types of signal sets. In particular, we obtain isometries between groups and geometrically uniform codes with a minimal generator. We also draw attention to the influence of the environment metric space, the group metric, and the matched mapping on the labeling of a signal set. The results are valid for all environment metric spaces. The alphabet emerges naturally from the relationship between the signal set S and the label group derived from its symmetry group, Γ(S).