Constrained Shell Mapping for Geometric Optimization of Multilevel 4D Modulations

In this work we demonstrate the design and numerical analysis of uncoded multilevel four-dimensional modulations defined by the constrained shell mapping over the discrete Hopf fibration. Employing a two-way map between spheres defined over respectively R4 and R3, the resulting modulations presents spatiotemporal distributions resembling both geometric and probabilistic shaping. To demonstrate effectiveness, some results are provided in the geometry and mutual information performance.