Sphere Shaping with Dispersion-Aware Sequence Selection for Long-Haul Transmission Systems

We introduce a nonlinear-tolerant design approach to Enumerative Sphere Shaping (ESS) for $M$-QAM signaling, utilizing sequence selection techniques based on a sign-independent metric: the Energy Dispersion Index, EDI (a scheme denoted E-ESS) and a novel sign-dependent metric: EDI of a dispersed sequence, D-EDI (a scheme denoted D-E-ESS). These approaches are designed to minimize rate loss and enhance transmission performance in nonlinear optical fiber transmission systems, catering to both short-distance and long-haul scenarios. Our simulation results reveal significant performance gains over conventional ESS, with improvements up to $0.4$~bits/4D-symbol. These improvements were observed over a 205 km single-span standard single mode fiber link in WDM transmission, with five dual-polarization channels, each operating at a net rate of $400$~Gbit/s. Furthermore, we demonstrate that D-E-ESS surpasses conventional ESS by $0.03$~bits/4D-symbol in achievable information rate over a $30$~span link with $80$~km span in a single-wavelength 8 discrete multi-bands (DMB) transmission with an $880$~Gbit/s net rate per channel, achieving similar performance to sequence selection based on a full split-step Fourier method (SSFM) simulation. We also demonstrate that D-EDI exhibits a more accurate negative correlation with the transmission performance compared to EDI. Notably, our proposed D-E-ESS scheme maintains robust performance improvements even when we consider reduced-complexity versions of it, consistently delivering throughput enhancements across various block lengths and selected sequence lengths.