On Product Codes with Probabilistic Amplitude Shaping for High-Throughput Fiber-Optic Systems

Alex Alvarado; Alexandre Graell i Amat; Alireza Sheikh

REVIEW 2 major objections 5 minor 27 references

On Product Codes with Probabilistic Amplitude Shaping for High-Throughput Fiber-Optic Systems

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper shows that probabilistic amplitude shaping can be combined with hard-decision-decoded product codes to gain up to 2.7 dB and one bit per channel use over uniform signaling in high-throughput fiber-optic systems.

desk verdict Solid incremental extension of PAS to product codes with a correct feasibility derivation, but the headline 2.7 dB/1 bpcu gains are not rigorously established because the uniform baseline comparison is under-specified and likely not rate-matched. read the letter →

arxiv 1908.04205 v4 pith:UT447QDE submitted 2019-08-09 cs.IT math.IT

Alireza Sheikh , Alexandre Graell i Amat , Alex Alvarado This is my paper · ORCID

classification cs.ITmath.IT

keywords probabilisticamplitudeshapingproductcodeshard-decisiondecodingiterativeboundeddistancefiber-opticcommunicationsspectralefficiencycodedmodulationstaircase

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Probabilistic amplitude shaping (PAS), which redistributes the probabilities of constellation points to approach the Shannon limit, has mostly been paired with soft-decision LDPC codes. This paper shows that PAS also works with product codes decoded by hard decisions, the low-complexity decoding choice for high-throughput fiber transponders, provided the component-code parameters obey a rate-matching feasibility condition and the amplitude and sign bits are spread evenly across the code array by a random interleaver. For 256-QAM over AWGN, the resulting scheme beats uniform signaling by up to 2.7 dB in signal-to-noise ratio and adds up to 1 bit per channel use of spectral efficiency at a block error rate of $10^{-3}$. The paper also demonstrates that a recently introduced decoder, iterative bounded-distance decoding with combined reliability (iBDD-CR), adds another 0.3 dB and lets product codes match or outperform staircase codes of the same rate.

What carries the argument

The central object is the PAS-PC encoding chain: a constant-composition distribution matcher (CCDM) generates shaped amplitudes; a random interleaver distributes the amplitude bits and the uniform sign/parity bits equally across all rows of the systematic product code array; and the systematically encoded parity bits are mapped to sign bits. The load-bearing identity is the rate balance $R = \tilde{k}^2/\tilde{n}^2 = (m-1+\gamma)/m$, which together with $n = \tilde{k}^2/(m-1+\gamma)$ converts the component-code shortening parameter $s$ into a discrete set of feasible spectral efficiencies $\gamma$; for $(v,t)=(10,3)$ and 16-ASK the paper finds 205 such values, versus 40 for staircase codes. The improvement mechanism in the second result is iBDD-CR decoding, which propagates channel-reliability information between row and column decoders.

What would settle it

Run the PAS-PC scheme with the tabulated parameters on a fiber testbed or split-step simulation with realistic launch powers; if the measured SNR gain over uniform signaling is substantially below 2.7 dB at $P_e=10^{-3}$, the AWGN modeling assumption is the point of failure. A second check: pick a shortened BCH component code with a shortening value $s$ that violates the integer-feasibility condition (6)-(8); if it still yields the designed spectral efficiency and target error rate, the paper's necessary conditions are not necessary.

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Extended reading notes

Core claim

The paper establishes that PAS is compatible with bit-wise hard-decision decoding of product codes, provided the component code parameters satisfy an integer-feasibility constraint derived from the PAS rate equation: for a PC with shortened BCH component code of length $\tilde{n}$ and information length $\tilde{k}$, the symbol block length $n = \tilde{k}^2/(m-1+\gamma)$ must be a positive integer and $\gamma n$ a non-negative integer. Over 256-QAM with a Maxwell-Boltzmann amplitude distribution, this yields operating points whose achievable rate $2H(A)+2\gamma$ approaches the shaped hard-decision information rate $R_{\mathrm{HDD}}$, and the simulated operating points show up to 2.7 dB SNR gain and about 1 bpcu spectral efficiency gain over uniform signaling at $P_e=10^{-3}$. With the iBDD-CR decoder, the same PAS-PC scheme gains up to 0.3 dB over plain iBDD and matches or beats staircase codes of equal rate at several shortening parameters.

Load-bearing premise

The fiber-optic channel is modeled as AWGN through the Gaussian noise model, so the 2.7 dB gain is computed for an idealized channel and may shrink under nonlinear fiber effects.

Editorial extensions

If this is right

PAS can be added to existing hard-decision product-code transceivers without changing the decoder, making shaping available to very high-throughput and low-latency fiber links.
The feasibility condition defines the achievable spectral efficiency grid for a given component-code family; product codes offer 205 rates versus 40 for staircase codes at $(v,t)=(10,3)$, enabling finer rate adaptation.
At a block error rate of $10^{-3}$, shaped PCs outperform uniform PCs by up to 2.7 dB and 1 bit/channel use, with the gain increasing at lower code rates.
Switching the decoder to iBDD-CR recovers up to 0.3 dB for the same PAS-PC setup and closes the performance gap to staircase codes at several rates.
The optimal operating point for each component code sits at the crossing of the achievable-rate curve $2H(A)+2\gamma$ with the shaped hard-decision information rate; at that point the simulated PC back-off is around 1.6 dB at $P_e=10^{-3}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same rate-balance argument should transfer to other product-like code families (e.g., braided BCH or half-product codes), so the feasibility condition may serve as a design rule beyond PCs and SCCs.
Since the shaping parameter $\lambda$ is optimized for the hard-decision achievable rate rather than for the hybrid decoder, the 0.3 dB iBDD-CR gain is likely conservative; re-optimizing for the hybrid metric could move the operating points further.
A hardware-oriented prediction: a deterministic interleaver that spreads bit levels uniformly, rather than a random permutation, should achieve the same shaping gain while saving memory, a testable implementation choice.
The comparison is made at $P_e=10^{-3}$; if deep-space or long-haul submarine links require $P_e$ near $10^{-9}$, the error floor of the shaped PC should be assessed before extrapolating the 2.7 dB gain.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Firas,

Quick read of the Sheikh et al. paper. The core contribution is straightforward: take the PAS scheme the group applied to staircase codes in [18], carry it over to product codes, and derive the necessary integrality conditions (Eqs. 5–8) for the BCH component code parameters. That derivation is clean and correct, and the observation that PCs give much finer SE granularity than SCCs (205 feasible s values versus 40) is a genuine practical point. The random interleaver for spreading bit levels across row/column codes is a sensible fix, and the iBDD-CR comparison is a nice extra. The paper is honestly written: the AWGN modeling assumption is flagged, and the conclusion admits that the AIR optimization for hybrid decoding is left open.

Where I part ways with the abstract is the 'up to 2.7 dB and 1 bpcu' headline. Those numbers come from Fig. 4, and the comparison protocol there is not explicit. The text says the PCs are chosen to have roughly the same code rate as the SCCs in [18], but it never says how the uniform baseline is rate-matched to each PAS operating point. Since PAS reduces SE relative to uniform signaling at the same code rate (H(A) < m-1), comparing PAS and uniform at the same s means comparing different SEs. To claim a 2.7 dB gain, you need to show the uniform PC at the same SE as the PAS point. The figure uses only eight s values for the uniform front, though the code parameters admit many more, so the envelope is coarse. The maximum horizontal gap could easily be an artifact of an under-sampled uniform curve. This is not an accusation of cooking the numbers, but the paper needs to pin down the comparison before those headline numbers are cited.

The other weakness is reproducibility: no code, no data, no error bars, and the random interleaver is only described as 'random', which doesn't allow someone to re-run the simulation. For a paper whose main evidence is simulation, that's a real gap.

Who should read this: people working on shaping with hard-decision BCH codes for optical transceivers. They will get the feasibility conditions and the PC vs SCC trade-off. It deserves peer review, but I would ask for a revision that spells out the baseline selection and ideally provides artifacts.

Referee Report

2 major / 5 minor

Summary. This paper applies probabilistic amplitude shaping (PAS) to product codes (PCs) with hard-decision decoding. In Section III, the authors derive integer-feasibility conditions (Eqs. (5) to (8)) linking the PC component-code shortening parameter s to the PAS tuning parameter gamma, and tabulate feasible parameters for v=10, t=3 in Table I. They propose a random interleaver to spread bit-levels across PC component codewords and apply both iterative bounded-distance decoding (iBDD) and iBDD with combined reliability (iBDD-CR). Section IV reports AWGN simulations at block-error probability 10^-3 for 256-QAM, comparing PAS with PCs to uniform signaling and to staircase codes (SCCs). The headline claims are gains up to 2.7 dB and spectral efficiency improvement up to about 1 bit/channel use over uniform signaling, plus up to 0.3 dB from iBDD-CR and a statement that PAS with PCs and iBDD-CR closes the gap to PAS with SCCs and iBDD.

Significance. The necessary-condition derivation is self-contained and does not rely on curve fitting; it gives a simple design rule that should transfer to other component-code parameters. The observation that PCs offer 205 feasible rate points versus 40 for SCCs is a concrete practical advantage. Extending iBDD-CR to PAS is a useful step, and the authors are transparent that their operating points are optimized for the Hamming-metric AIR, so the iBDD-CR gains are conservative. If the headline gains survive a rate-matched comparison and the simulation details are made reproducible, this would be a solid contribution to high-throughput coded modulation with hard-decision decoding. In its current form, the numerical claims are not yet independently verifiable from the manuscript.

major comments (2)

[Section IV, Fig. 4] The headline claim of up to 2.7 dB gain and 1 bpcu SE improvement is not supported by a clearly defined comparison. The text does not state how the uniform-signaling baseline is matched to each PAS operating point in terms of spectral efficiency or code rate. For a fixed component code (fixed s), uniform signaling has SE 2(m-1+gamma), while PAS has SE 2H(A)+2gamma with H(A)<m-1, so the uniform baseline has higher SE at the same s; any gain at fixed SE must come from comparing PAS at one code rate with uniform at a different code rate. The uniform envelope in Fig. 4 is built from only eight shortening values (s=77, 289, 447, 605, 661, 727, 759, 797), although uniform PCs allow many more shortenings (they are not restricted by Eq. (8)). The paper also does not specify how the two envelopes are interpolated between the simulated points. As a result, the reported maxima may occur at a SE or SNR where no uniform code was simulated, and the 2.7 dB / 1 bpcu figures may be inflated by code-rate granularity rather than by shaping. Please define the comparison protocol precisely, e.g., rate-match the uniform code to each PAS operating point (nearest feasible rates above and below), and report gains for matched pairs; or explicitly state the envelope interpolation and the sensitivity of the maxima to the baseline code set.
[Sections III and IV] The numerical results cannot be independently verified from the manuscript. The random interleaver that distributes the bit-levels among PC component codewords is described only as 'a random interleaver' with no construction, seed, or permutation length; since the authors argue this interleaver is necessary for symmetric PCs, the simulation outcomes depend on an unspecified implementation. In addition, no code, data, or error bars are provided, and the number of simulated blocks needed to estimate the 10^-3 block-error probability is not stated. Please specify the interleaver (e.g., a fixed permutation generated by a described pseudo-random process), the number of trials, and the uncertainty in the reported operating SNRs, or make the simulation code/data available.

minor comments (5)

[Section IV, Fig. 5 caption] The caption lists PC shortening values '77, 266, 447, 535, 605'; elsewhere in the text and Fig. 3 the second value is 261. Please correct the typo.
[Section IV] The SCC shortening parameters are inconsistent: the text gives '63, 274, 431, 519, 591' in one place, '63, 271, 431, 591, 647, 711, 743, 783' in another, and the Fig. 5 caption gives '63, 247, 431, 519, 591'. Please unify these lists.
[Section II] In the description of the PAS encoder, 'The vector u is parsed to u^{gamma n} and u^k' should read 'parsed into'.
[Eq. (9)] The sets S_l^0 and S_l^1 are described as 'sets of size 2^m ASK symbols'; since the two sets partition the 2^m-ary ASK alphabet by one bit-level, each should have size 2^{m-1}. Please check the typesetting.
[Abstract and Conclusion] The phrases 'up to 2.7 dB' and 'up to 1 bpcu' are not qualified in the abstract/conclusion; Section IV notes that only 5 of 205 feasible operating points were simulated in Fig. 3 and 8 in Fig. 4. Please state explicitly that these maxima are over the tested subset of component codes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central PAS-with-PC gains come from direct Monte Carlo simulation, and the self-cited decoder and SCC baselines are independent published results.

full rationale

The paper's headline gains (2.7 dB and 1 bpcu) are obtained by simulating the actual PAS+PC encoders/decoders in Sec. IV and comparing them with simulated uniform-signaling PCs and with SCC baselines from [18]. No parameter appearing in the derivation is fitted to these reported gains. The feasibility conditions for PC component codes are derived algebraically from the rate identity R=(m-1+gamma)/m and the integer constraints in Eqs. (3)-(8); they do not assume the target result. The Maxwell-Boltzmann shaping parameter is optimized per SNR by maximizing the achievable information rate R_HDD in Eq. (2), which is an independent design criterion, not a fit to the simulated block-error-rate points. The iBDD-CR decoder from [6] and the SCC baseline from [18] are prior published results by the same authors, but they are used as external building blocks/comparison points and are not invoked as unverified premises that force the conclusion; the paper's contribution is the application of PAS to PCs and the corresponding decoder comparison, which is evaluated by simulation. The skeptical concern that the 2.7 dB/1 bpcu envelope comparison in Fig. 4 may be affected by an under-sampled or not rate-matched uniform baseline is a fairness/validity question, not a circularity one: the reported numbers are not equivalent to the inputs by construction, and no equation in the paper reduces the claimed gain to a fitted parameter or to the definition of the baseline. Therefore the derivation chain is self-contained and non-circular.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central derivation relies on the PAS architecture, the AWGN channel assumption, and the R_HDD achievable rate from prior literature. The only free parameter is the shaping parameter λ, which is optimized per SNR. No new physical entities are introduced.

free parameters (1)

λ (shaping parameter) = optimized per SNR via Eq. (2)
The Maxwell-Boltzmann shaping parameter is tuned to maximize R_HDD for each SNR; it is a design parameter rather than a fitted-to-target value, but it affects all reported operating points.

assumptions (4)

domain assumption The fiber-optic channel is modeled as an AWGN channel via the Gaussian noise model.
Footnote 2 states this modeling assumption; all reported SNR gains are computed under this model and may not hold for nonlinear fiber channels.
standard math The achievable information rate R_HDD for bit-wise HDD from [22, eq. (62)] is valid.
Used in Eq. (2) to optimize the shaping parameter λ and to define feasible SNR regions; accepted from prior literature without derivation.
domain assumption The Maxwell-Boltzmann distribution with parameter λ is the shaping distribution, and the optimal λ is found by maximizing R_HDD.
The paper restricts shaping to the MB family following [16], [18]; no proof is given that this family is optimal for the HDD metric.
domain assumption The iBDD-CR decoding algorithm [6] achieves the performance claimed for the hybrid decoder.
The paper relies on the algorithm introduced in a self-cited arXiv preprint [6] and does not re-derive its behavior; this supports the 0.3 dB gain claim.

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Cite this review

Pith. "Pith review of On Product Codes with Probabilistic Amplitude Shaping for High-Throughput Fiber-Optic Systems." pith.science (2026). https://pith.science/paper/UT447QDE

@misc{pith2026190804205,
  author       = {Pith},
  title        = {Pith review of: On Product Codes with Probabilistic Amplitude Shaping for High-Throughput Fiber-Optic Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UT447QDE}},
  note         = {Machine review of arXiv:1908.04205}
}

abstract

Probabilistic amplitude shaping (PAS) can flexibly vary the spectral efficiency (SE) of fiber-optic systems. In this paper, we demonstrate the application of PAS to bit-wise hard decision decoding (HDD) of product codes (PCs) by finding the necessary conditions to select the PC component codes. We show that PAS with PCs and HDD yields gains up to $2.7$ dB and SE improvement up to approximately $1$ bit/channel use compared to using PCs with uniform signaling and HDD. Furthermore, we employ the recently introduced iterative bounded distance decoding with combined reliability of PCs to improve performance of PAS with PCs and HDD.

Figures

Figures reproduced from arXiv: 1908.04205 by the authors.

**Figure 1.** Block diagram of the CM scheme with PAS and PC under consideration. The parameters shown in the system model corresponds to encoding [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Code array of a PC. Table I: Parameters of the designed PCs for v = 10, t = 3, and 16-ASK modulation γ 0.7682 0.7503 0.6912 0.6797 0.5942 0.5233 0.4464 0.3645 0.2303 0.1426 0.00854 s 3 77 261 289 447 535 605 661 727 759 797 n˜ 1020 946 762 734 576 488 418 362 296 264 226 ˜k 990 916 732 704 546 458 388 332 266 234 196 n 260100 223729 145161 134689 82944 59536 43681 32761 21904 17424 12769 γn 199800 167869 100341 9154… view at source ↗

**Figure 4.** Simulation results of PAS with PC and SCCs with parameters [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: Performance of the PAS with PC and iBDD-CR algorithm, PC with [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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