  Probabilistic amplitude shaping (PAS) combines an outer shaping layer with an
inner, systematic forward error correction (FEC) layer to close the shaping
gap. Proposed for PAS, constant composition distribution matching (CCDM)
produces amplitude sequences with a fixed empirical distribution. We show that
CCDM suffers from high rate losses for small block lengths, and we propose to
use Enumerative Sphere Shaping (ESS) instead. ESS minimizes the rate loss at
any block length. Furthermore, we discuss the computational complexity of ESS
and demonstrate that it is significantly smaller than shell mapping (SM), which
is another method to perform sphere shaping. We then study the choice of design
parameters for PAS. Following Wachsmann et al., we show that for a given
constellation and target rate, there is an optimum balance between the FEC code
rate and the entropy of the Maxwell-Boltzmann distribution that minimizes the
gap-to-capacity. Moreover, we demonstrate how to utilize the non-systematic
convolutional code from IEEE 802.11 in PAS. Simulations over the additive white
Gaussian noise (AWGN) and frequency-selective channels exhibit that ESS is up
to 1.6 and 0.7 dB more energy-efficient than uniform signaling at block lengths
as small as 96 symbols, respectively, with convolutional and low-density
parity-check (LDPC) codes.
