  In this paper, we study amplitude shaping schemes for the probabilistic
amplitude shaping (PAS) framework as well as algorithms for
constant-composition distribution matching (CCDM). Huffman-coded sphere shaping
(HCSS) is discussed in detail, which internally uses Huffman coding to
determine the composition to be used and relies on conventional CCDM algorithms
for mapping and demapping. Numerical simulations show that HCSS closes the
performance gap between distribution matching schemes and sphere shaping
techniques such as enumerative sphere shaping (ESS). HCSS is based on an
architecture that is different from the trellis-based setup of ESS. It allows
to tailor the used HCSS compositions to the transmission channel and to take
into account complexity constraints. We further discuss in detail multiset
ranking (MR) and subset ranking (SR) as alternatives to arithmetic-coding (AC)
CCDM. The advantage of MR over AC is that it requires less sequential
operations for mapping. SR operates on binary alphabets only, which can
introduce some additional rate loss when a nonbinary-to-binary transformation
is required. However, the binomial coefficients required for SR can be
precomputed and stored in a lookup table (LUT). We perform an analysis of rate
loss and decoding performance for the proposed techniques and compare them to
other prominent amplitude shaping schemes. For medium to long block lengths,
MR-HCSS and SR-HCSS are shown to have similar performance to ESS. SR-HCSS and
uniform 64QAM are compared in additive white Gaussian noise simulations and
shaping gains of 0.5 dB and 1 dB are demonstrated with 1 kbit and 100 kbit LUT
size, respectively.
