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We propose Four-Dimensional (4D) energy limit enumerative sphere shaping(ESS) of $M$-QAM signaling to minimize rate loss and improve the transmissionperformance over non-linear WDM optical-fiber systems. Simulation results showthat the proposed scheme outperforms the conventional ESS by$0.19$~bit/4D-symbol in achievable information rate over a $205$-km single-spanlink and a WDM transmission of five polarization-division-multiplexed channelswith $400$-Gbit/s net rate per channel. We also study the achieved performanceover several shaping block lengths and show that the achieved gains do notscale well over multi-span systems.

Multi-dimensional Energy Limitation in Sphere Shaping for Nonlinear Interference Noise Mitigation