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Coset Shaping for Coded Modulation

Irina Bocharova, Maiara F. Bollauf, Boris Kudryashov

TL;DR

The paper introduces coset shaping, a new shaping technique for coded $2^m$-PAM/QAM that can shape both information and parity bits without added encoder/decoder complexity. By combining a high-rate shaping code with a coset-based selection of low-energy representatives, the scheme achieves a tunable energy distribution and approaches the AWGN capacity as the code length $n$ and modulation order $M=2^m$ grow, under suitable assumptions. An asymptotic analysis based on random coset codes and shaping regions shows the rate can approach capacity, with the shaping region NSM $G(\Lambda)$ converging to $1/(2\pi e)$. Simulation results for long NB QC-LDPC codes indicate that coset shaping outperforms or matches existing shaping methods (e.g., PAS) in practical scenarios, particularly in the error-floor region, validating its efficiency and practicality for coded modulation.

Abstract

A new shaping technique called coset shaping for coded QAM and PAM signaling is introduced and analyzed. This technique can be applied not only to information bits but also to parity bits without incurring additional complexity costs. It is proven that as the length of the error-correcting code and the modulation order tend to infinity, the gap to capacity for the proposed shaping scheme can be made arbitrarily small. Numerical results and comparisons for the shaping scheme, along with nonbinary LDPC-coded QAM signaling, are presented.

Coset Shaping for Coded Modulation

TL;DR

The paper introduces coset shaping, a new shaping technique for coded -PAM/QAM that can shape both information and parity bits without added encoder/decoder complexity. By combining a high-rate shaping code with a coset-based selection of low-energy representatives, the scheme achieves a tunable energy distribution and approaches the AWGN capacity as the code length and modulation order grow, under suitable assumptions. An asymptotic analysis based on random coset codes and shaping regions shows the rate can approach capacity, with the shaping region NSM converging to . Simulation results for long NB QC-LDPC codes indicate that coset shaping outperforms or matches existing shaping methods (e.g., PAS) in practical scenarios, particularly in the error-floor region, validating its efficiency and practicality for coded modulation.

Abstract

A new shaping technique called coset shaping for coded QAM and PAM signaling is introduced and analyzed. This technique can be applied not only to information bits but also to parity bits without incurring additional complexity costs. It is proven that as the length of the error-correcting code and the modulation order tend to infinity, the gap to capacity for the proposed shaping scheme can be made arbitrarily small. Numerical results and comparisons for the shaping scheme, along with nonbinary LDPC-coded QAM signaling, are presented.
Paper Structure (8 sections, 5 theorems, 9 equations, 5 figures, 3 tables)

This paper contains 8 sections, 5 theorems, 9 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Modulation and Mapping for Unshaped Coded PAM
  5. Coset shaping
  6. Asymptotic analysis
  7. Simulation results
  8. Conclusion

Key Result

Theorem 1

In the product ensemble $\CMcal{C}_{\textnormal{c}} \times \CMcal{P}$, for large enough $m$ and $n$, there exists a shaping construction, such that an arbitrarily small error probability in AWGN channel with parameters $(P,\sigma^2)$ can be achieved if the code rate per signal dimension $R=k/n_s$ sa where $G(\Lambda)$ is the NSM of $\Lambda$, and $o(n)\to 0$ when $n\to \infty$.

Figures (5)

  • Figure 1: Shaping scheme
  • Figure 2: Shaping-oriented form of the code generator matrix, $k_s=0$.
  • Figure 3: Two-dimensional shaped Gray-coded modulation. The sixty-four black dots show 64-QAM points. The eight signal points corresponding to the eight codewords are represented by two coset images, one by red squares in Fig. A, and the other by red circles in Fig. B. Each signal point is labeled by the message bits. The non-linear sphere shaper in Fig. D chooses four minimum-energy points among all eight signal points.
  • Figure 4: Mapping of codewords to PAM signals: a) Example \ref{['ex2']}, b) Example \ref{['ex3']}
  • Figure 5: Comparison of NB LDPC coded QAM-256 signaling with and without shaping with li2025coded, $R_{\textnormal{c}}=5.33$ bits per QAM signal, Shannon limit is 15.97 dB, BICM limit is 17.02 dB, QAM limit is 15.99 dB

Theorems & Definitions (17)

  • Example 1
  • Definition 1
  • Example 2
  • Example 3
  • Definition 2
  • Definition 3
  • Theorem 1
  • Proposition 1
  • proof
  • Lemma 1
  • ...and 7 more