{"id":"f8861e86-7b71-4777-b754-d8da179f6244","arxiv_id":"2506.01761","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A 32-point QAM constellation and binary labeling, optimized for relative-intensity-noise-limited IM-DD channels, improve SNR by 0.94 dB over cross QAM-32 and push BER toward the Gray-label limit.","lead":"This paper designs a new 2D symbol constellation for short-range fiber links that use pulse-amplitude modulation, choosing the 32 best points out of 36 to avoid symbol-dependent laser noise. The design gives a 0.94 dB signal-to-noise gain over the standard cross QAM-32 constellation and lowers error rates in the noise-dominated regime.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The global optimality of X*_32 in Eq. (8) is not established: the search covers only 345 diagonal-symmetric subsets, and channel symmetry alone does not guarantee a symmetric minimizer among all 58,905 subsets.","rationale":"The reader's weakest_assumption identifies exactly the same gap we consider load-bearing: the exhaustive search is restricted to diagonal-symmetric subsets, and no proof is given that a globally optimal 32-point subset must be symmetric. Our independent analysis confirms that the channel symmetry only pairs each subset with its coordinate-swapped image; it does not imply existence of a symmetric optimal subset. The 345 count is correct, and the SNR equations and labeling heuristic appear internally consistent. The conclusion's 'minimizes BER' phrasing is also stronger than the body's conjecture, but that is secondary and already noted by the reader. Because the practical 0.94 dB gain and the near-Gray BER behavior are not invalidated by the missing optimality proof, the appropriate verdict remains CONDITIONAL: the paper should either prove or explicitly replace the global-optimality claim. If the proposed exhaustive test reveals an asymmetric subset with lower SER, the verdict should move to REJECT for the optimality claim, though the heuristic contribution may still stand.","tokens_in":6024,"tokens_out":7152,"duration_ms":84298,"concrete_test":"At one floor-regime operating point (e.g., RIN=-144 dB/Hz, OMA=4 dBm), enumerate all 58,905 subsets of X36 and evaluate SER under the ML rule (5) using the same Monte Carlo or numerical-integration procedure as in Fig. 3. If any asymmetric subset has lower SER than X*_32, the global-optimality claim of Eq. (8) is false; if none does, the symmetry reduction is validated for that regime. If full enumeration is too costly, use a branch-and-bound search over subsets with a certified lower bound, or at minimum enumerate all subsets obtained by replacing the four removed points of X*_32 with other excluded-point combinations and their coordinate swaps.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the reduction in Eq. (8). The paper minimizes SER over all 58,905 subsets of X36, but the actual search is restricted to the 345 subsets invariant under (Y1,Y2)->(Y2,Y1), justified only by 'the symmetry of the noise distribution.' The channel has the symmetry p(y|x)=p(y^T|x^T), so any subset and its coordinate-swapped image have equal SER. That does not prove that the set of minimizers contains a symmetric subset; asymmetric minimizers could come in equal-performance pairs, and no averaging argument yields a valid 32-point symmetric subset with the same SER. Thus the claim that X*_32 solves (8) over all subsets, and the conclusion that it offers the lowest SER among 32-point PAM-6 constellations, are unsupported. The demonstrated 0.94 dB SNR gain over X+_32 and the near-SER/5 BER behavior in the tested cases are not called into question by this gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Felipe Villenas et al. propose a 32-point 2D constellation X*_32, a subset of the 6x6 QAM-36 grid, for PAM-6 based IM-DD links dominated by relative intensity noise. They formulate a minimum-SER search over all 32-point subsets of QAM-36, reduce the search to the 345 subsets symmetric about the Y1=Y2 diagonal, and report a constellation with four removed points in the first quadrant. For this constellation they design a quasi-Gray binary labeling. They compare the proposed constellation with cross QAM-32 and a reference QAM-32 from [9] via analytic SNR evaluation and Monte Carlo SER/BER simulations, reporting a 0.94 dB SNR gain over cross QAM-32 in the saturation regime and BER that approaches SER/5 asymptotically.","tokens_in":6179,"tokens_out":5907,"duration_ms":60960,"significance":"If the optimality claim can be rigorously established, the paper provides a simple and useful design principle for RIN-limited IM-DD: remove symbols where both coordinates are large positive, thereby reducing the signal-dependent noise variance. The 0.94 dB SNR gain is modest but relevant for short-reach links. The exhaustive search over the 345 symmetric subsets is transparent and reproducible, and the labeling search over 10! candidates for the remaining points is a concrete optimization step. However, the global optimality claim and the asymptotic optimality of the labeling are stronger than the evidence presented. The paper is clearly written and the Monte Carlo methodology is standard.","major_comments":[{"comment":"The paper states in Eq. (8) that X*_32 minimizes SER over all 58,905 subsets of QAM-36, but the actual search only covers the 345 subsets that are invariant under coordinate swap (Y1,Y2)->(Y2,Y1). The noise distribution in Eq. (4) is symmetric in the two coordinates, so any subset and its transposed image have equal SER; however, this does not imply that the set of minimizers contains a symmetric subset. The SER objective is not convex, and mirror-image minimizers could exist as an asymmetric pair without a symmetric representative. Therefore, the claim that X*_32 achieves the lowest SER among all 32-point PAM-6 constellations is not established. The authors should either prove existence of a symmetric minimizer, perform the full search over 58,905 subsets, or explicitly restrict the claim to the symmetric-subset search.","section":"Optimizing QAM-32 Constellation, Eq. (8)"},{"comment":"The phrase 'the OMA region of interest' is never defined. The noise variance in Eq. (1) depends on eta through sigma_th^2 = (NEP/eta)^2 B, and OMA = eta(max X - min X), so the SER in (8) depends on OMA. As a consequence, the optimal subset could vary with OMA. Please specify the OMA range used in the optimization and verify that X*_32 is optimal over that range, or state explicitly that the optimization is performed in the asymptotic RIN-dominated limit where sigma_th = 0.","section":"Optimizing QAM-32 Constellation"},{"comment":"The conclusion states that the proposed labeling is 'asymptotically optimal' and that 'any improvements by jointly optimizing the constellation X32 and the labeling will be negligible.' These statements are stronger than the evidence. The labeling section describes a heuristic three-step construction and presents a conjecture, not a proof. I recommend replacing 'asymptotically optimal' with 'approaching the Gray-bound SER/5 in the simulated error-floor region' and removing the claim about negligible improvement from joint optimization.","section":"Conclusions"}],"minor_comments":[{"comment":"The Monte Carlo SER/BER results do not report the number of simulated symbols or confidence intervals. At BER levels around 10^-7, this information is necessary to assess whether the apparent gains are statistically significant; please add it.","section":"Results, Fig. 3(b-c)"},{"comment":"The description 'The six adjacent points are then labeled' is ambiguous; please state which points are adjacent to the initial 16-point subset and how their labels are chosen.","section":"Optimizing QAM-32 Constellation, Step 2"},{"comment":"The horizontal lines are labeled 'SER/5' in the figure and called 'approximated Gray label BER values' in the text; clarify that these lines represent the approximation BER = SER/log2(M).","section":"Results, Fig. 3(c)"},{"comment":"The abstract uses 'PAM-6' as a 1D modulation, but the transmitted symbols are actually 2D points from a 32-point subset of QAM-36. Consider rephrasing to avoid the impression that the proposal is a 1D constellation.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper is well suited to a short or workshop-style venue, but for a journal the gap between the stated global optimality and the actual symmetric-subset search is a load-bearing issue that must be fixed. The 0.94 dB gain claim is likely robust and should not be lost in revision. The authors may also check whether the 'first optimization' claim is compatible with prior art in [9] and related work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this is a well-executed paper on a narrow but practical problem, and the main quantitative result, a 0.94 dB SNR gain over cross QAM-32 in a RIN-dominated IM-DD link, is credible. The paper's one serious overreach is the claim that the proposed constellation is the SER-minimizing 32-point subset of QAM-36. The search only covers the 345 subsets symmetric about the Y2=Y1 diagonal, and channel symmetry alone does not guarantee that a global minimizer is symmetric; asymmetric minimizers could come in equal-performance pairs. So the 'lowest SER' phrasing in the conclusion is not supported. That is a real flaw, but it is a flaw in an optional claim: the demonstrated gain over the two existing baselines does not depend on global optimality.\n\nWhat is genuinely new: the asymmetric subset that removes the four high-noise first-quadrant points is a concrete, sensible design I have not seen before. The combinatorial count checks out (58,905 total, 345 symmetric), and the SNR expressions are internally consistent. The labeling heuristic is reasonable, and the BER curves approaching SER/5 in the error floor are a nice practical result. The paper also does the right thing by computing everything, including Monte Carlo simulations, from a stated channel model rather than fitting to a target.\n\nMinor issues worth mentioning: the 'OMA region of interest' is never defined, and the optimal subset could in principle depend on OMA. The conclusions say the labeling 'minimizes the BER,' while the body only conjectures optimality; those should be aligned. And the Monte Carlo curves have no error bars, which is common for this venue but still worth noting.\n\nWho should read it: anyone working on PAM-6 for 400G-per-lane IM-DD links, and people doing constellation selection with signal-dependent noise. The paper deserves a serious referee; it is a solid engineering contribution that needs a caveat, not a rewrite. I would ask the authors to either prove the symmetry reduction or soften the global-optimality claim, specify the OMA region, and release their simulation parameters. With those changes, I would be happy to cite it.","headline":"Solid, useful constellation-design paper; the 0.94 dB gain is real, but the global optimality claim rests on an unproven symmetry reduction.","tokens_in":6764,"tokens_out":2544,"would_cite":true,"duration_ms":25032,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proposes a 32-point two-dimensional constellation, selected from the $6\\times 6$ PAM-6 grid, that achieves the lowest symbol error rate among its symmetric 32-point subsets under relative-intensity-noise-dominated IM-DD…","keywords":["relative intensity noise","IM-DD","PAM-6","QAM-32 constellation design","symbol error rate","bit labeling","Gray coding","optical interconnects"],"falsifier":"Re-run the search without the diagonal-symmetry restriction, over all 58,905 subsets of QAM-36 using the same ML detection rule, or evaluate a concrete asymmetric candidate that removes two high-noise symbols from each of two quadrants rather than four from one quadrant, at RIN = $-141$ dB/Hz and high OMA, and compare its SER with $\\mathcal{X}^\\star_{32}$; if any asymmetric subset has lower SER, the claimed optimality of $\\mathcal{X}^\\star_{32}$ is refuted.","tokens_in":5787,"feed_emoji":"📡","tokens_out":9669,"duration_ms":80431,"temperature":0.7,"pith_summary":"The paper proposes a new 5-bit/2D-symbol modulation format for intensity-modulation direct-detection links whose performance is limited by laser relative intensity noise (RIN). It claims that, among all 32-point subsets of the square $6\\times 6$ PAM-6 grid, the constellation $\\mathcal{X}^\\star_{32}$ obtained by exhaustive search has the lowest symbol error rate in the RIN-dominated regime, and that it gains 0.94 dB in signal-to-noise ratio over the conventional cross QAM-32. The paper also designs a binary labeling for $\\mathcal{X}^\\star_{32}$ whose bit error rate approaches $\\mathrm{SER}/5$ at high received power, which is the Gray-code limit even though the constellation itself cannot be Gray-labeled. If correct, this gives a drop-in constellation change for next-generation 400 Gb/s-per-lane links, improving margin or relaxing laser RIN requirements.","feed_headline":"RIN-optimized 32-QAM beats cross QAM-32 by 0.94 dB","feed_subtitle":"A 5-bit/2D-symbol format, designed for laser relative-intensity noise, also attains Gray-like BER in the error floor.","key_machinery":"The engine of the design is the memoryless IM-DD channel model $Y = X + Z\\sqrt{\\sigma_{\\mathrm{th}}^2 + (X+\\beta)^2\\sigma_{\\mathrm{rin}}^2}$, in which the total noise variance depends on the transmitted symbol, so RIN makes high-amplitude symbols noisier. The optimization problem is (8): minimize SER over the 58,905 subsets $X_{32}\\subset X_{36}$, reduced to 345 candidates by restricting to constellations invariant under the diagonal symmetry of the RIN noise distribution. The winning constellation $\\mathcal{X}^\\star_{32}$ removes four symbols in the first quadrant, where the sum of the two noise variances is largest. The labeling is a three-step heuristic: a Gray-coded 16-point square base, a new most-significant bit, and a search over $10!$ labelings of the remaining points minimizing average Hamming distance between nearest neighbors.","core_discovery":"The central claim is that the constellation $\\mathcal{X}^\\star_{32}$, found by minimizing symbol error rate over all 32-point subsets of QAM-36 that are symmetric about the $Y_2=Y_1$ diagonal, is the SER-minimizing 32-point subset for the RIN-dominated IM-DD channel. Unlike cross QAM-32, which drops the four corners, $\\mathcal{X}^\\star_{32}$ removes four points from the first quadrant, where the signal-dependent RIN noise is largest, and is therefore not symmetric about its origin. In the OMA region where RIN dominates, $\\mathcal{X}^\\star_{32}$ improves SNR by 0.94 dB over cross QAM-32, while the reference constellation of [9] improves by 0.71 dB over the same baseline; the corresponding SER and BER floors are also reduced. The paper argues that its heuristic labeling makes the BER approach $\\mathrm{SER}/5$ asymptotically, the Gray-label limit for 32 points, although two nearest-neighbor pairs have Hamming distance 3.","pith_inferences":["If the symmetry restriction is removed, a full search over all 58,905 subsets might find an asymmetric constellation with slightly lower SER; the 0.94 dB gain over cross QAM-32 would likely survive, but the 'lowest-SER' claim would need revision.","The same subset-selection principle, dropping symbols in the quadrant with largest RIN variance, should carry over to PAM-8 or 2D formats with more bits per symbol, since the scalar noise model becomes even more lopsided at higher amplitudes.","The labeling is heuristic; a joint constellation-and-labeling optimization, even within the symmetric family, could reduce the Hamming distance of the two exceptional nearest-neighbor pairs and push BER even closer to $\\mathrm{SER}/5$.","Using the SNR definition in (6) as an objective for geometric shaping, with continuous symbol positions instead of a fixed grid subset, may yield gains beyond the 0.94 dB."],"forward_implications":["In a 400 Gb/s-per-lane IM-DD link, replacing cross QAM-32 with $\\mathcal{X}^\\star_{32}$ gives a 0.94 dB SNR gain in the RIN-limited regime, which can be spent as extra link margin or as a relaxed RIN specification for the laser.","Because the proposed labeling achieves BER approximately equal to $\\mathrm{SER}/5$ at high OMA, forward error correction designed for Gray-coded QAM-32 can be used without extra overhead.","The optimization method of (8) with the diagonal-symmetry reduction applies directly to other operating points, such as different RIN values or different bias $\\beta$, and to other subset sizes.","The result shows that for signal-dependent noise, the best 2D constellation for a fixed alphabet is not the one that maximizes minimum distance; asymmetric subset selection is beneficial."],"supporting_citations":[{"why":"Provides the RIN model with $\\sigma_{\\mathrm{rin}}^2 = 10^{\\mathrm{RIN}/10} B$, which defines the symbol-dependent noise in the channel.","marker":"[4]"},{"why":"Defines the reference constellation $X_{32}^{\\mathrm{ref}}$ and its Gray labeling, the main intermediate baseline in the comparison.","marker":"[9]"},{"why":"Supplies the same system setup and parameter values used for the Monte Carlo SER and BER simulations.","marker":"[11]"},{"why":"Provides the quasi-symmetric ultracomposite labeling used for the cross QAM-32 baseline in the BER comparison.","marker":"[13]"},{"why":"Supplies the Gray-coding theory that identifies $\\mathrm{SER}/\\log_2(32)$ as the asymptotic BER limit that the proposed labeling approaches.","marker":"[16]"}],"fun_headline_variants":["RIN-tuned 32-QAM subset gains 0.94 dB over cross QAM-32","New 5-bit/2D format trims RIN noise for 0.94 dB boost","Asymmetric 32-point QAM design edges cross QAM-32 by 0.94 dB","RIN-aware constellation cuts error floor, nears Gray limit","PAM-6-based 32-point format beats cross QAM-32 in RIN channel"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the globally best 32-point subset of QAM-36 is symmetric about the diagonal noise axis; the exhaustive search only examines the 345 symmetric candidates, and no proof is given that an asymmetric subset could not have a lower symbol error rate.","fun_headline_variants_meta":{"raw":{"variants":["RIN-tuned 32-QAM subset gains 0.94 dB over cross QAM-32","New 5-bit/2D format trims RIN noise for 0.94 dB boost","Asymmetric 32-point QAM design edges cross QAM-32 by 0.94 dB","RIN-aware constellation cuts error floor, nears Gray limit","PAM-6-based 32-point format beats cross QAM-32 in RIN channel"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000336,"raw_usage":{"total_tokens":1798,"prompt_tokens":819,"completion_tokens":979,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":435,"completion_tokens_details":{"reasoning_tokens":861}},"tokens_in":435,"tokens_out":979,"duration_ms":8295,"temperature":1.0,"reasoning_tokens":861,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:34:17.023806+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the search without the diagonal-symmetry restriction, over all 58,905 subsets of QAM-36 using the same ML detection rule, or evaluate a concrete asymmetric candidate that removes two high-noise symbols from each of two quadrants rather than four from one quadrant, at RIN = $-141$ dB/Hz and high OMA, and compare its SER with $\\mathcal{X}^\\star_{32}$; if any asymmetric subset has lower SER, the claimed optimality of $\\mathcal{X}^\\star_{32}$ is refuted.","supporting_citations":[{"cited_title":"4-PAM for high-speed short-range optical communications","cited_arxiv_id":null,"evidence_quote":"Provides the RIN model with $\\sigma_{\\mathrm{rin}}^2 = 10^{\\mathrm{RIN}/10} B$, which defines the symbol-dependent noise in the channel."},{"cited_title":"PAM-6 coded modula- tion for IM/DD channels with a peak-power constraint","cited_arxiv_id":null,"evidence_quote":"Defines the reference constellation $X_{32}^{\\mathrm{ref}}$ and its Gray labeling, the main intermediate baseline in the comparison."},{"cited_title":"On Geometric Shaping for 400 Gbps IM-DD Links with Laser Intensity Noise","cited_arxiv_id":null,"evidence_quote":"Supplies the same system setup and parameter values used for the Monte Carlo SER and BER simulations."},{"cited_title":"Constellation labeling for linear encoders","cited_arxiv_id":null,"evidence_quote":"Provides the quasi-symmetric ultracomposite labeling used for the cross QAM-32 baseline in the BER comparison."},{"cited_title":"Gray coding for multilevel constellations in Gaussian noise","cited_arxiv_id":null,"evidence_quote":"Supplies the Gray-coding theory that identifies $\\mathrm{SER}/\\log_2(32)$ as the asymptotic BER limit that the proposed labeling approaches."}],"review_version":1}