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

Abstract

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).

Editorial of Special Issue “Optimization and Applications of Modern Wireless Networks and Symmetry”

“…This book contains the successfully published submissions [1][2][3][4][5][6] to a Special Issue of Symmetry on the subject area of "Optimization and Applications of Modern Wireless Networks and Symmetry".…”

“…Despite emphasizing the hyperbolic case because of its rich properties and Euclidean cases for historical reasons and applications, the results are valid for any metric space. Additionally, the article opens up new possibilities for applying discrete metrics that come from the group metric obtained from the (finitely generated) group of labels [4].…”