Soft Demapping of Spherical Codes From Cartesian Powers of PAM Constellations

For applications in concatenated coding for optical communications systems, we examine soft-demapping 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 demapping 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 demapping with frozen symbols provides a gain of about 0.3 dB in signal-to-noise ratio compared to the traditional symbol-by-symbol demapping. 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 demapping 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. Overall, using the spherical codes containing all possible permutation codes of the same average power and the BCJR algorithm, a gain of 0.5 dB is observed compared with the case of using one permutation code with the symbol-by-symbol demapping. Comparison of the achievable information rates of bit-metric decoding verifies the observed gains.