An approximate way to calculate the confidence scores used in optical-link decoding matched an exact calculation in simulations, while requiring fewer operations as the modulation constellation grows, according to a preprint. The proposed shortcut, called a zero-crossing approximation, or ZCA, matched the exact method's reported error rates after error correction for the tested links. A simpler AWGN-like calculation, by contrast, produced higher residual error levels in some coded tests.
How the shortcut works
The study asks whether a piecewise-linear approximation of exact log-likelihood ratios, or LLRs, can preserve soft-decision decoding for Gray-coded PAM-4 and PAM-8. LLRs are numerical confidence scores passed to a decoder; ZCA builds them from the points where the exact score crosses zero and from the exact slopes at those points. The derivation uses a max-log approximation, but the practical test is whether the lower-complexity score changes the result after decoding.
The simulated channel was an intensity-modulation/direct-detection link limited by relative intensity noise, or RIN. In the model, RIN was Gaussian, but each PAM level had its own noise distribution. The comparison therefore put exact LLRs beside an AWGN-like mismatched calculation and the ZCA shortcut.
The first decoding tests
To compare the methods at the decoder output, the researchers used Monte Carlo channel observations with a (128, 120) extended-Hamming inner code. Their Chase-II decoder focused on the three least-reliable bits, flipped up to two of them, and examined seven test patterns in total. The setup produced bit-error-rate comparisons after error correction for the exact, AWGN-like and ZCA calculations.
At 600 Gb/s, the AWGN-like error floor was 7.2 times the exact-LLR result. At 800 Gb/s, it was 3.5 times higher. ZCA matched the exact method's post-FEC BER at both rates in the reported simulations.
PAM-8 also exposed a limit of the code being tested. Under the reported RIN conditions, its extended-Hamming BER did not reach the KP4-FEC threshold, although ZCA still matched exact LLRs. The authors say that setting needs a stronger error-correcting code, so matching the metric did not change the reported code outcome.
Sensitivity to imperfect conditions
ZCA's advantage was not unconditional. With low noise-variance estimation errors, it outperformed the AWGN-like LLRs up to a normalized error standard deviation of 14%. Below 10%, it virtually preserved that improvement. When the errors became large, the AWGN-like calculation was favored instead.
Bandwidth restrictions changed the size of the gap, but did not separate ZCA from the exact method. In bandwidth-limited simulations, ZCA matched exact-LLR BER. The difference between that pair and the AWGN-like result narrowed as the limitation worsened, then disappeared in the 90 GHz case.
Information rate tells a more mixed story
An achievable-information-rate analysis told a similar, but not identical, story. At low optical modulation amplitude, ZCA was below matched generalized mutual information, or GMI, then approached it as the amplitude increased. The AWGN-like mismatched rate diverged downward and was 0.0081 below the maximum in the reported comparison. That distinction matters because the study found that GMI was not a reliable predictor of every finite-code error floor.
For PAM-8, the normalized information score NGMI saturated around 0.93 under the reported RIN conditions. At a symbol rate of 238.13 GBd, the corresponding maximum achievable information rate was 664.38 Gb/s, compared with 714.39 Gb/s in the comparison case. Across the RIN sweep, ZCA and exact LLRs reached the same maximum NGMI, while the AWGN-like penalty grew to a gap of 0.028. PAM-8 saturation began above -150 dB/Hz, while PAM-4 saturation appeared around -142 dB/Hz or higher.
A second code kept the pattern
The authors also ran LDPC simulations using 64,800-bit code blocks, improved min-sum decoding and a 238.13 GBd symbol rate. Across the three metrics, post-FEC BER versus NGMI was essentially identical. But the AWGN-like method showed an OMA gap of 0.1 dB versus exact LLRs at code rate 3/4, 0.36 dB at 5/6 and 2.41 dB at 8/9. ZCA overlaid the exact results, while the AWGN-like BER showed an early error floor below the KP4 threshold.
What remains untested
The findings remain tied to the simulated channel and code cases described here. The comparison covered Gray-coded PAM-4 and PAM-8, extended-Hamming and LDPC decoding, and the reported RIN and bandwidth scenarios. It does not establish performance for other formats, shaping methods, codes or channel conditions. No physical transmission validation is reported, and the manuscript is labeled a preprint.
The paper reports funding from Holland High Tech | TKI HSTM through the PPS allowance scheme for public-private partnerships for the COCOLI project.
Paper data and sources
Original title: Low-complexity Soft-decision LLR Calculations for Next-generation IM-DD Systems with RIN
Authors: Felipe Villenas, Yunus Can Gültekin, Alex Alvarado
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text