Detection Limits of Power-Domain NOMA: Information-Theoretic Bounds and Spatial Diversity

Reproducibility code for the paper

Liang Dong and Robert W. Heath Jr., "On the Detection Limits of Power-Domain NOMA: Information-Theoretic Bounds and Spatial Diversity," IEEE Transactions on Communications, vol. 74, pp. 13995-14011, 2026, doi: 10.1109/TCOMM.2026.3726732.

The paper develops an information-theoretic framework for the detection limits of power-domain NOMA under finite-alphabet modulation. It identifies the two roles of power allocation (setting the SIC decoding order and shaping the composite-constellation geometry), derives the constellation-constrained mutual-information hierarchy $I_{\mathrm{TIN}} \le I_{\mathrm{MAP}} \le I_{\mathrm{oracle}} \le \log_2 M$ with the structural-gain and oracle-deficit metrics, establishes Fano-type lower and pairwise-error-probability upper bounds on the symbol error rate, and extends the analysis to the MIMO-NOMA uplink, where a regime transition at $N_r = K$ governs whether linear MMSE-SIC approaches the MAP bound. This repository contains the simulation code that produces every numerical result and figure in the manuscript.

Contents

File	Role
sim_learned_mud.py	Base SISO downlink NOMA library: constellations, Rayleigh channel, marginal-MAP / oracle / TIN detectors, mutual-information kernels, IEEE plotting style
sim_mimo_noma.py	Base MIMO uplink NOMA library: spatial channel, MMSE / MMSE-SIC / MAP / oracle detectors
sim_it_bounds.py	SISO information-theoretic bounds: CCMI hierarchy, structural gain and oracle deficit, Fano/PEP SER bounds, composite minimum-distance geometry, power-differentiation sweep
sim_mimo_it_bounds.py	MIMO information-theoretic bounds: MI hierarchy, SER, MAP-oracle gap vs. spatial diversity, the $N_r = K$ regime transition
sim_revision1.py	First-round revision additions: imperfect-CSI GMI, per-user fairness, and the power-allocation universality sweep across $K$ and modulation
sim_revision2.py	Second-round revision additions: the nearest-neighbor multiplicity factor (deterministic), MIMO-NOMA under spatially correlated receive channels, and an expectation-propagation (EP) approximate-MAP detector
replot_it_bounds.py	Regenerate the SISO figures from cached results, without re-running the simulation
replot_mimo.py	Regenerate the MIMO figures from cached results
replot_revision1.py	Regenerate the first-round revision figures from cached results
replot_revision2.py	Regenerate the second-round revision figures from cached results
sim_it_bounds_results.pkl	Cached SISO Monte Carlo results (MI/SER vs. SNR, power sweep)
sim_mimo_results.pkl	Cached MIMO Monte Carlo results (MI/SER vs. SNR for $N_r \in {1,2,3,4}$)
revision1_results.json	Cached imperfect-CSI, fairness, and power-universality results
sim_revision2_corr.pkl	Cached correlated-channel MI sweep (MI hierarchy vs. SNR for $N_r \in {3,4}$, receive correlation $r \in {0, 0.7, 0.9}$)
sim_revision2_ep.pkl	Cached EP approximate-MAP SER ($N_r = 2$)

sim_it_bounds.py and sim_revision1.py import sim_learned_mud; sim_mimo_it_bounds.py imports sim_mimo_noma; and sim_revision2.py imports both sim_mimo_noma and sim_mimo_it_bounds. All base/analysis modules are bundled here, so the repository is self-contained.

Requirements

• Python 3.9 or later
• NumPy, SciPy, Matplotlib, PyTorch (CUDA optional but strongly recommended)

pip install -r requirements.txt

Quick start

# SISO bounds: CCMI hierarchy, information gaps, Fano/PEP SER bounds, power sweep
python sim_it_bounds.py

# MIMO bounds: MI hierarchy, SER, MAP-oracle gap, regime transition at N_r = K
python sim_mimo_it_bounds.py

# First-round revision additions: imperfect CSI, multi-user fairness, power-allocation universality
python sim_revision1.py

# Second-round revision additions: multiplicity factor, correlated channels, EP approximate-MAP detector
python sim_revision2.py

The cached .pkl / .json results are included, so the figures can be regenerated in seconds without a GPU:

python replot_it_bounds.py
python replot_mimo.py
python replot_revision1.py
python replot_revision2.py

Reproducibility notes

• All experiments use a fixed seed (SEED = 42), three users with the default power allocation P = [0.2, 0.3, 0.5], i.i.d. Rayleigh block fading with unit variance, and a reference SNR of 16 dB for cross-comparisons.
• The SISO and MIMO Monte Carlo runs use 4,000,000 samples per SNR point; the revision runs use 2,000,000 (MI/SER) and 500,000 (power sweep) samples.
• The power-differentiation sweep of sim_it_bounds.py (Fig. "power_mi") and the $K=3$ curve of the universality sweep in sim_revision1.py share the identical allocation path through the default [0.2, 0.3, 0.5], so they coincide to within Monte Carlo error.
• Figures are written as EPS and PDF with Type-42 (TrueType) fonts for IEEE production. They are not tracked in the repository (see .gitignore); run the replot_*.py scripts to regenerate them.
• The 16-QAM, $K=4$ marginal-MAP case enumerates $16^4 = 65536$ composite hypotheses and is skipped on an 8 GB GPU.
• The second-round additions (sim_revision2.py) compute the nearest-neighbor multiplicity factor deterministically by enumerating the composite constellation (no Monte Carlo), use 1,500,000 samples per SNR point for the correlated-channel MI sweep, and 4,000,000 for the EP approximate-MAP SER. Receive correlation uses the exponential model $[\mathbf{R}]_{mn} = r^{|m-n|}$, and the EP detector runs 10 iterations with damping 0.2. The EP run reuses the same seeds as sim_mimo_results.pkl, so the MMSE-SIC / MAP / Oracle curves in the regenerated SER figure are identical to those of sim_mimo_it_bounds.py.

Citation

@article{dong2026detection_limits,
  author  = {Liang Dong and Robert W. {Heath Jr.}},
  title   = {On the Detection Limits of Power-Domain {NOMA}:
             Information-Theoretic Bounds and Spatial Diversity},
  journal = {IEEE Trans. Commun.},
  volume  = {74},
  pages   = {13995--14011},
  year    = {2026},
  doi     = {10.1109/TCOMM.2026.3726732}
}

The companion paper on coded NOMA achievable rates is coded-noma-finite-alphabet, published as IEEE Trans. Commun., vol. 74, pp. 13392-13407, 2026, doi: 10.1109/TCOMM.2026.3726750.

License

Released under the MIT License. See LICENSE.