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In this paper we carry out a joint optimization of probabilistic (PS) andgeometric shaping (GS) for four-dimensional (4D) modulation formats inlong-haul coherent wavelength division multiplexed (WDM) optical fibercommunications using an auto-encoder framework. We propose a 4D 10 bits/symbolconstellation which we obtained via end-to-end deep learning over thesplit-step Fourier model of the fiber channel. The constellation achieved 13.6%reach increase at a data rate of approximately 400 Gbits/second in comparisonto the ubiquitously employed polarization multiplexed 32-QAM format at aforward error correction overhead of 20%.

High-Cardinality Hybrid Shaping for 4D Modulation Formats in Optical Communications Optimized via End-to-End Learning