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.