Geometric shaping: low-density coding of Gaussian-like constellations

Constellation shaping is necessary to approach channel capacity for information rates above 1 bit/dim. Probabilistic shaping shows a small gap to capacity, however a complex distribution matcher is required to modify the source distribution. Spherical shaping of lattice constellations also reduces the gap to capacity, but practical Voronoi shaping is feasible in small dimensions only. In this paper, our codebook is a real geometrically non-uniform Gaussian-like constellation. We prove that this discrete codebook achieves channel capacity when the number of points goes to infinity. Then we build a special mapping to interface between non-binary low-density codes and the codebook, allowing the code alphabet size to be equal to the square root of the codebook size. Excellent performance is shown with fast-encoding and practical iterative probabilistic decoding, e.g. 0.7 dB gap to capacity at 6 bits/s/Hz with a code defined over the ring Z/8Z.