  We propose a 4-dimensional 2-ary amplitude ring-switched modulation format
with 64 symbols, which is denoted as 4D-2A-RS64 encoded over two polarization
tributaries to improve the transmission performance over long-haul optical
fibers in the presence of the non-linear Kerr effect. At a spectral efficiency
of 6 bits per 4D, simulation results show that this format outperforms the
polarization division multiplexed (PDM) 8QAM-star modulation as well as the
4D-2A-8PSK over links without inline dispersion management. We evaluate the
performance for a WDM transmission of $11\times90~\mathrm{Gbaud}$ channels over
a multi-span SSMF link. For an achievable information rate of
$4.8\mathrm{bit/s/Hz}$, the maximum transmission distance is improved by
$10.6\%$ (400 km) and $4\%$ (160 km) compared to PDM-8QAM-star and 4D-2A-8PSK
respectively. The achieved gains are composed of a linear part and a non-linear
part, respectively from the improved Euclidean-distance distribution and the
constant power property of the 4D modulation. The geometric shaping of the
proposed scheme is easy to implement and is robust to Mach-Zehnder modulator
(MZM) imbalances and quantization errors stemming from the finite
digital-to-analog converter (DAC) resolution. This robustness is compared to
the one of other geometric-shaped non-linearity tolerant 4D schemes such as the
4D-2A-8PSK and the 4D-64PRS that can be both outperformed by our scheme in
severe conditions.
