  For applications in concatenated coding for optical communications systems,
we examine the encoding and soft-decoding of short spherical codes constructed
as constant-energy shells of the Cartesian power of pulse amplitude modulation
constellations. These are unions of permutation codes having the same average
power. We construct a list decoder for permutation codes by adapting Murty's
algorithm, which is then used to determine mutual information curves for these
permutation codes. In the process, we discover a straightforward expression for
determining the likelihood of large subcodes of permutation codes. We refer to
these subcodes, obtained by all possible sign flips of a given permutation
codeword, as orbits. We introduce a simple process, which we call orbit
decoding with frozen symbols, that allows us to extract soft information from
noisy permutation codewords. In a sample communication system with
probabilistic amplitude shaping protected by a standard low-density
parity-check code that employs short permutation codes, we demonstrate that
orbit decoding with frozen symbols provides a gain of about 0.3 dB in
signal-to-noise ratio compared to the traditional symbol-by-symbol decoding. By
using spherical codes composed of unions of permutation codes, we can increase
the input entropy compared to using permutation codes alone. In one scheme, we
consider a union of a small number of permutation codes. In this case, orbit
decoding with frozen symbols provides about 0.2 dB gain compared to the
traditional method. In another scheme, we use all possible permutations to form
a spherical code that exhibits a computationally feasible trellis
representation. The soft information obtained using the BCJR algorithm
outperforms the traditional symbol-by-symbol method by 0.1 dB.
