  Approaching Shannon's capacity via geometric shaping has usually been
regarded as challenging due to modulation and demodulation complexity,
requiring look-up tables to store the constellation points and constellation
bit labeling. To overcome these challenges, in this paper, we study
lattice-based geometrically shaped modulation formats in multidimensional
Euclidean space. We describe and evaluate fast and low complexity modulation
and demodulation algorithms that make these modulation formats practical, even
with extremely high constellation sizes with more than $10^{28}$ points. The
uncoded bit error rate performance of these constellations is compared with the
conventional QAM formats in the additive white Gaussian noise and nonlinear
fiber channels. At a spectral efficiency of 2 bits/sym/polarization, compared
with 4-QAM format, transmission reach improvement of more than 38% is shown at
the hard-decision forward error correction threshold of $2.26\times 10^{-4}$.
