Geometric and Probabilistic Shaping to Enable LDPC Encoding with Linear Complexity

Previous examples of probabilistic amplitude shaping (PAS) for low-density parity-check (LDPC) codes use protographs that lack the structure that facilitates low-complexity encoding. This paper optimizes the signaling positions and probability mass function (PMF), i.e. we use both geometric and probabilistic shaping, to facilitate LDPC encoders with encoding complexity that is linear in blocklength while still harvesting the capacity-approaching benefits of PAS. The new approach equalizes the conditional entropies of the bits that label the constellation points, which decouples the LDPC design from the specific signaling PMF. An example shows how the new signaling PMF and a newly designed protograph-based LDPC code with rate 2/3 and blocklength n = 64, 800 allow encoding with linear complexity while slightly outperforming previous designs by operating within 0.57 dB of the additive white Gaussian noise channel capacity at FER 10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−3</sup> with a low-complexity encoder.