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Soft Demapping of Spherical Codes from Cartesian Powers of PAM Constellations

Reza Rafie Borujeny, Susanna E. Rumsey, Stark C. Draper, Frank R. Kschischang

TL;DR

A list decoder for permutation codes is constructed by adapting Murty’s algorithm, which is then used to determine mutual information curves for these permutation codes, and a straightforward expression for determining the likelihood of large subcodes of permutation codes is discovered.

Abstract

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 called orbits. We introduce a simple process, called orbit demapping, 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 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 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. 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. Comparison of the achievable information rates of bit-metric decoding verifies the observed gains.

Soft Demapping of Spherical Codes from Cartesian Powers of PAM Constellations

TL;DR

A list decoder for permutation codes is constructed by adapting Murty’s algorithm, which is then used to determine mutual information curves for these permutation codes, and a straightforward expression for determining the likelihood of large subcodes of permutation codes is discovered.

Abstract

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 called orbits. We introduce a simple process, called orbit demapping, 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 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 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. 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. Comparison of the achievable information rates of bit-metric decoding verifies the observed gains.
Paper Structure (28 sections, 1 theorem, 57 equations, 14 figures, 1 table, 3 algorithms)

This paper contains 28 sections, 1 theorem, 57 equations, 14 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Permutation Codes
  4. Probabilistic Amplitude Shaping
  5. The Assignment Problem
  6. Permutation Codes as Multidimensional Constellations
  7. List Decoding of Variant I Permutation Codes
  8. Orbit Decoding of Variant II Permutation Codes
  9. Soft Demapping of Permutation Codes
  10. Variant I Encoder Function and its Inverse
  11. Variant II Encoder Function and its Inverse
  12. Permutation Encoding for PAS
  13. Soft-in Soft-out Demapping
  14. Numerical Evaluation
  15. Structure of Constant-Energy Shells of PAM Constellations
  16. ...and 13 more sections

Key Result

Theorem 1

The likelihood of the orbit of $\bm{x}\in \mathcal{C}$ under the action of $\phi$ is given by where $a(\mathtt{E}_{\bm{y}}, \mathtt{E}, \sigma, n)$ is a constant that depends on the energy of the received word $\mathtt{E}_{\bm{y}}$, the energy $\mathtt{E}$ of each permutation codeword, the noise standard deviation $\sigma$ and the blocklength $n$.

Figures (14)

  • Figure 1: Block diagram of the communication system that we consider.
  • Figure 2: Mutual information curves for five codes with $n=12$.
  • Figure 3: Mutual information curves for five codes with $n=50$.
  • Figure 4: BMD rates for various spherical codes with $n=50$ and $\mathtt{E}=530$ with a $8$-PAM constituent constellation are shown.
  • Figure 5: Block error rates for various spherical codes with $n=50$ and $\mathtt{E}=530$ with a $8$-PAM constituent constellation. The $(10\,860, 8\,448)$ LDPC code from the 5G NR standard is used in the PAS architecture.
  • ...and 9 more figures

Theorems & Definitions (19)

  • Theorem 1
  • proof
  • Example 1
  • Example 2
  • Remark 1
  • Example 3
  • Example 4
  • Definition 1
  • Definition 2
  • Example 5
  • ...and 9 more