Nonbinary LDPC QAM Signaling

• Nonbinary LDPC-coded QAM signaling is a modulation approach that combines NB-LDPC codes with high-order QAM to boost spectral efficiency and improve error-floor performance.
• It employs joint iterative detection-decoding algorithms and optimized mapping strategies like SICM to reduce decoding complexity while achieving significant performance gains.
• Advances such as probabilistic shaping, FFT-based decoding, and multilevel coding extend its application to large-scale MIMO and high-dimensional communication systems.

Nonbinary LDPC-coded QAM signaling refers to coded modulation systems in which nonbinary low-density parity-check (NB-LDPC) codes defined over finite fields GF($q$), with $q>2$, are combined with high-order quadrature amplitude modulation (QAM) constellations. This paradigm unifies the strengths of NB-LDPC coding—improved minimum distance, better error-floor behavior, more efficient algebraic structure—with constellation choices that maximize spectral efficiency, such as 16-QAM, 64-QAM, or higher. The NB-LDPC code’s parity-check matrix is typically sparse and constructed over GF($q$), enabling direct symbol mapping and iterative joint detection-decoding. Recent advances encompass joint detection-decoding algorithms such as IJDD, multilevel coding architectures, probabilistic shaping strategies, constellation mapping optimization, and specialized approaches for high-dimensional scenarios such as large-scale MIMO. The overarching theme is performance gains of up to 1 dB or higher over binary-coded counterparts, together with dramatic reductions in decoding complexity via algorithmic innovations.

1. System Architecture: Coding, Mapping, and Modulation

The transmitter generates source symbols $\mathbf{u}\in$ GF($q$)$^K$, encodes them to codeword $\mathbf{v}\in\mathcal{C}[N,K]\subset$ GF($q$)$^N$ using an LDPC code defined by a sparse parity-check matrix $\mathbf{H}\in$ GF($q$)$^{M\times N}$, $M=N-K$ (Wang et al., 2010). Each symbol $v_j$ is mapped via a bijection $\mathcal{M}:$ GF($q$) $\to\mathcal{X}$ where $\mathcal{X}$ is an $M$-QAM constellation with $|\mathcal{X}|=q$ (Wang et al., 2010, Potapova et al., 2017).

Channel observation follows $y_j = x_j + n_j$, $n_j\sim\mathcal{CN}(0, N_0)$, where $x_j=\mathcal{M}(v_j)$. At the receiver, $y_j$ is either (i) demapped and detected via maximum likelihood (ML) or soft-distance metrics, or (ii) further processed for probabilistic shaping (Boutros et al., 2017). Nonbinary codes enable direct symbol-to-signal mapping, avoiding the inefficiencies and error propagation of bit-interleaved coded modulation under high-order modulations.

2. Decoding Algorithms and Complexity

2.1. Joint Detection-Decoding (IJDD)

The IJDD algorithm alternates between ML hard-decision detection and message-passing decoding using exclusively hard messages. ML detection computes:

$$\hat{x}_j^{(k)} = \arg\min_{x\in\mathcal{X}} \|y_j^{(k)} - x\|$$

followed by syndroming and extrinsic-check updates in GF($q$), variable-node voting, and feedback correction:

$$y_j^{(k+1)} = y_j^{(k)} + \xi_j^{(k)}\,\vec{L}_j^{(k)}$$

with step-size $\xi_j^{(k)}$ determined by voting statistics and reliability threshold $T$ (Wang et al., 2010). This denoising feedback accelerates convergence at low complexity.

2.2. FFT-QSPA and Multilevel Coding

Traditional $q$-ary sum-product algorithms (QSPA) propagate soft messages of length-$q$ per edge, incurring $O(M\,d_c\,q\log q)$ real operations per iteration. Multilevel coding (MLC) decomposes the $m=\log_2 M$ bits of each QAM symbol into $L$ levels, each protected by an NB-LDPC code over a smaller field GF($2^{m'}$), drastically reducing per-iteration complexity (Potapova et al., 2017). FFT-based CN updates and sequential level-wise decoding deliver near-optimal performance with ∼$4\times$ lower complexity versus full-field decoding.

2.3. Belief Propagation and EXIT Chart Optimization

NB-BP approaches use vector-message passing for M-ary symbol detection and EXIT chart-driven irregular LDPC profile selection to maximize performance in large-scale MIMO with high-dimensional constellations (Narasimhan et al., 2013). Complexity is $O(KN\sqrt{M})$ per iteration with no matrix inversion.

3. Constellation Mapping Strategies

Performance depends critically on symbol-to-constellation mapping. Ungerboeck set-partitioning splits $M$-QAM into cosets maximizing minimum Euclidean distance for MSBs. In optimized symbol-interleaved CM (SICM), mapping groups high-significance bits into amplitude for uniform reliability; amplitude-sign CM (ASCM) and bit-plane CM (BPCM) offer control over reliability hierarchy (Bocharova et al., 2021).

Probabilistic shaping adapts the symbol distribution to approach the Maxwell–Boltzmann energy profile, exploiting distribution matchers (CCDM) for desired symbol probabilities. Full shaping on circular QAM (CQAM) distributing $p^2$ points over $p$ shells with $p$ phases is realizable with nonbinary codes over prime fields GF($p$) (Boutros et al., 2017).

4. Performance Benchmarks and Analytical Bounds

Representative results show NB-LDPC codes over GF($q$), matched via optimized mapping, achieve 0.5–1 dB decoding threshold gains over binary codes for 16-QAM, 64-QAM, or 256-QAM (Bocharova et al., 2021, Potapova et al., 2017). For instance, SICM mapping for GF(16)/16-QAM attains FER $10^{-4}$ at SNR 10.0 dB versus 10.6 dB for binary mapping; for GF(64)/64-QAM, gains are similar.

Multilevel NB-LDPC coding over GF(16) attains the same waterfall and error-floor behavior as full-field GF(64)/GF(256) with up to $4\times$ lower complexity (Potapova et al., 2017). Random coding ML bounds via ensemble-average Euclidean distance spectra track BP simulation curves within 0.2–0.6 dB for FER $<10^{-4}$ (Bocharova et al., 2021).

Probabilistically shaped NB-LDPC–coded CQAM achieves gaps of 0.09–0.1 dB to AWGN capacity at $R_c=2/3$, outperforming classical time-sharing PAS by ∼0.5 dB in shaping gain (Boutros et al., 2017).

5. Implementation Guidelines and Practical Considerations

Quantize received samples $y_j^{(k)}$ in fixed-point (≥8 bits) and correction steps via simple arithmetic (Wang et al., 2010, Potapova et al., 2017). Set the radius $r=1.415\,d_{\min}$ and vote threshold $T=3$. Parallel or blockwise updating is feasible for hardware acceleration; message memory is minimized by hard-decision and compact representation.

Select field size so that GF($2^{m'}$) effectively protects critical bits. Level rates $R_i$ should satisfy overall spectral efficiency constraints and mitigate error floors. For probabilistic shaping, tune the Maxwell–Boltzmann parameter via grid search and set CQAM geometry parameters (e.g., $\rho_{\max}$, $\beta$) to maximize minimum distance (Boutros et al., 2017).

In MIMO, use NB-BP detection over $\sqrt{M}$-PAM alphabet with $40$ BP iterations, $O(KN\sqrt{M})$ complexity, and optimized degree profiles for selected loading factors $\alpha$ (Narasimhan et al., 2013).

6. Comparative Analysis: Binary vs. Nonbinary LDPC-Coded QAM

NB-LDPC codes matched to QAM constellations via optimized mapping consistently outperform binary LDPC codes in both waterfall and error-floor regimes. For practical standards such as WiFi (802.11n) and 5G NR, NB QC-LDPC codes over GF(16) and GF(64) under SICM mapping produce gains of 0.5–0.7 dB at similar complexity (Bocharova et al., 2021).

Complexity is tractable for $q=16,32,64$ using fast Hadamard transform, extended min-sum, or FFT-convolution techniques. Multilevel architectures offer near-field performance with substantial reductions in operations and memory.

7. Extension to Shaped and High-Dimensional Constellations

Probabilistic shaping and full CQAM architectures further narrow the gap to Shannon capacity, with nonbinary codes enabling direct control of amplitude distribution and joint symbol mapping (Boutros et al., 2017). These designs generalize to large QAM, APSK, and non-AWGN channels, with per-level channel state information and protograph-based code structure. In large-scale MIMO, nonbinary belief propagation achieves strong SNR gains over linear detectors, with practical deployment via EXIT chart code optimization (Narasimhan et al., 2013).

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In summary, nonbinary LDPC-coded QAM signaling, when implemented with careful symbol-to-constellation mapping, joint iterative detection-decoding, probabilistic shaping, and multilevel coding, offers significant improvements in both performance and hardware scalability for high-dimensional communications systems (Wang et al., 2010, Potapova et al., 2017, Boutros et al., 2017, Bocharova et al., 2021, Narasimhan et al., 2013).