Circular QAM with Sphere Shaping and Non-binary LDPC Codes in the Finite Length Regime

In this paper, we show how to combine non-binary low-density parity-check (LDPC) codes over $\mathbb{F}_{q}$ with enumerative sphere shaping (ESS). Our work combines LDPC codes with non-uniform signaling of new geometrically shaped circular quadrature amplitude modulation (CQAM) that reaches the Gaussian capacity. Furthermore, we show numerical results of the frame error rate (FER) for short blocklengths and the advantage yielded by shaping compared to the uniform signaling.