{"id":"31c0ec0a-1cd5-4e5d-bf34-0e8dc2c342a7","arxiv_id":"2506.06796","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A polarization-based finite-field multiple-access code with two decoders is shown by simulation to beat polar random spreading by about 1.25 dB for 15 users on a Gaussian multiple-access channel.","lead":"This paper designs a new type of polar code for multiple users sending data at the same time over a shared wireless channel. The authors report that their scheme needs about 1.25 dB less power than a leading alternative for 15 users, which could matter for low-power machine-type communications.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reported 1.25 dB gain depends on an internally inconsistent power/rate normalization (Eq. 15 vs §II.C–D and §VI.B); without a corrected simulation the comparison to polar random spreading is not reproducible.","rationale":"The reader's weakest_assumption concerns the transfer of the polarizing construction from ideal SC/capacity proxy to SCL/TopL-BMD decoders. That is a legitimate risk, but it is conditional on first having a well-defined simulation configuration. The more basic problem is that the configuration reported in §VI.B is not internally consistent: the encoder's information dimension M is fixed to JK in §II.C–D, yet §VI.B keeps M=992 while varying J, and Eq. (15) uses K instead of JK in the power budget. For the headline J=15, K=32 case, this changes R from 544 to 32 and leaves 512 information positions as permanently zero, which changes both spectral efficiency and energy per active symbol relative to the baseline. No amount of decoder transfer will fix that. I therefore flag the normalization inconsistency as the single load-bearing concern, and the proposed re-simulation with consistent M, R, and power constraint settles it. The reader's conditional verdict remains appropriate, so I did not change the verdict.","tokens_in":20690,"tokens_out":14640,"duration_ms":171604,"concrete_test":"Reproduce the J=15, K=32 BER curve after enforcing M=JK and R=m−M (hence M=480, R=544) and replacing Eq. (15) with mPavg = JK·μinfPavg + R·μredPavg, keeping total energy per information bit identical to the polar-spreading baseline. Also plot the paper's stated M=992, R=32 configuration under the same total energy. If the current 1.25 dB advantage at BER=10^-5 is not reproduced under the corrected normalization, the headline gain is an artifact; if it persists, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing weakness is the parameter/power normalization of the headline simulation. The encoder defined in §II.C–II.E fixes the systematic information-block length at M=JK (M=K×J), with parity length R=m−M. The power constraint Eq. (15) uses only K·μinf, not JK·μinf, and Section VI.B reports M=992 (R=32) for every J=5,15,31 with K=32. For J=15 this gives JK=480, so either M should be 480 and R=544, or 512 of the 992 information positions are permanently zero while still entering the power/rate budget. The baseline polar code (64,32) plus length-16 spreading occupies all 1024 degrees of freedom with the same 480 information bits. Because Eb/N0 and the reported coding gain depend on total energy per information bit and on how many zero positions are transmitted, the 1.25 dB gain is not well defined as written. This is not a side issue: if the normalization is corrected, the gain may shrink or disappear.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a polarized element-pair (EP) code for finite-field multiple-access (FFMA) systems over a Gaussian multiple-access channel (GMAC). A systematic encoder partitions the codeword into information and parity sections, a capacity expression (Eq. 28) is used to justify power allocation, and two decoders are introduced (SCL for balanced payloads and TopL-BMD for small payloads). The central claimed result is a BER gain over polar random spreading, e.g., 1.25 dB at BER 10^-5 for J=15, K=32 with m=1024, M=992, CRC=8, L=512 (Section VI.B). The main evidence is simulation, with parameters described in Section VI.","tokens_in":20982,"tokens_out":15613,"duration_ms":149008,"significance":"If the reported gains survive a consistent normalization, the paper offers a practical finite-blocklength multiuser coding scheme with lower decoding complexity than the polar random spreading baseline; the TopL-BMD decoder is an interesting and potentially useful algorithmic contribution. The manuscript specifies concrete simulation parameters and makes falsifiable BER predictions, which is a strength. However, the analytical support is weak: the capacity formula is cited from the authors' earlier work, the power normalization is internally inconsistent, and the code construction depends on an ideal-SC Monte Carlo proxy whose transfer to the actual decoders is not validated. The significance therefore hinges on whether the simulation claims can be reproduced under a corrected power/rate budget.","major_comments":[{"comment":"Section II.D defines the systematic information section by M = K × J, but Section VI.B sets m = 1024, K = 32, M = 992, and R = 32 for J = 5, 15, and 31. Only J = 31 satisfies M = J K; for J = 15, J K = 480, so either the systematic encoder should use M = 480 and R = 544, or the 512 positions beyond J K are frozen zeros whose status in the rate and power budgets is never specified. Because Eb/N0 and the claimed 1.25 dB gain depend on the energy per information bit and on the number of active positions, the simulation results in Fig. 9 are not reproducible as written.","section":"Section II.D vs. Section VI.B"},{"comment":"Equation (15) states m P_avg = K μ_inf P_avg + R μ_red P_avg and calls this 'total transmit power', but the capacity expression in Eq. (28) counts J K information symbols and an R-symbol parity section with J-user superposition. If Eq. (15) is a per-user constraint, the phrase 'total transmit power' is wrong and the comparison to polar spreading (which transmits all J users over the same m degrees of freedom) needs an explicit total-power normalization; if it is a total-power constraint, a factor J is missing from the right-hand side. Either way, the Monte Carlo search for μ_pas in Section VI.A and the resulting BER curves in Figs. 8 and 9 are not tied to a well-defined power budget.","section":"Eq. (15) vs. Eq. (28)"},{"comment":"The abstract and contribution list claim a derivation of the channel capacity, but Eq. (28) is quoted from the authors' earlier preprint [42]; the surrounding derivation is not a proof (the BI-ASC capacity in Eq. (22) is a BSC approximation and Eq. (27) is an asymptotic limit). Since the code construction in Section IV.B and the optimal power allocation in Section VI.A are both keyed to Eq. (28), the paper's analytical claims should be either re-derived here or explicitly presented as inherited from [42] with its assumptions restated.","section":"Section IV.A.3, Eq. (28)"},{"comment":"Section IV.B constructs the polarized index set A by Monte Carlo capacities of an ideal SC decoder, but the systems evaluated in Section VI use SCL and TopL-BMD decoders. The paper provides no finite-blocklength evidence, beyond the final BER curves, that the index set and μ_pas selected under the ideal-SC proxy remain near-optimal for the actual decoders; this is a load-bearing premise for the reported gains, and a sensitivity analysis (e.g., comparing constructions designed for each decoder) is needed.","section":"Sections IV.B and VI"}],"minor_comments":[{"comment":"The manuscript contains many typos and spelling errors ('ogranized', 'polaried', 'virous', 'matrx', 'Krnonecker', 'Capactiy', 'Suppse', 'blcok', 'anlayze'); a careful proofreading pass is required.","section":"Throughout"},{"comment":"The notation 'arg max p(y_i)' is undefined and appears twice; the capacity should be maximized over the input distribution, not the output distribution.","section":"Eq. (22)"},{"comment":"The product formula for P(w = ŵ | y) assumes independent bit posteriors and does not account for the CRC or the code structure; the notation uses M in the product while w has length J K. Please clarify the derivation or state the independence assumption explicitly.","section":"Lemma 1, Eq. (31)"},{"comment":"The text says 'which is half of the payload in Fig. 9' when discussing the K = 32 results, but the preceding paragraph for K = 64 references Fig. 8; the figure numbering and cross-references should be reconciled.","section":"Section VI.B"},{"comment":"The 'Marto Loco method' mentioned in the abstract is not defined or referenced in the body; please give the formal name and a citation, or remove the term.","section":"Abstract and body"},{"comment":"The baseline description specifies 'random Gaussian spreading sequences' but does not provide a seed or a precise MMSE-SCL-SIC implementation; please add enough detail (or a pseudo-code block) to make the comparison reproducible.","section":"Section VI.B"}],"recommendation":"major_revision","confidential_remarks":"The core capacity formula and several design elements come from the authors' own unpublished arXiv preprints [40]-[42], and the unresolved normalization issue in Sections II.D and VI.B means the headline gain may be an artifact. If the normalization is corrected and the simulations remain favorable, the paper could be suitable for publication; at present, I cannot recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper has a genuinely interesting idea—systematic polar codes built for FFMA on the GMAC—and the TopL-BMD decoder is a nice addition. But the headline 1.25 dB gain over polar random spreading is not reproducible as written, because the simulation settings contradict the system model on a fundamental parameter.\n\nThe system model says M = K×J, so with K=32 and J=15 the information section should be 480 bits and the parity section 544. The simulations use M=992 (R=32) for every J, meaning 512 information positions are permanently zero for J=15. The power constraint (15) counts only K nonzero information positions per user, while the capacity formula (28) counts JK. Which one is it? The Eb/N0 comparison and the coding gain depend on the answer. If the code is actually rate 480/(480+544) rather than the implied ultra-high rate, the gain could shrink or vanish. This is not a typo—it is the load-bearing assumption behind the simulation.\n\nWhat the paper does well: the systematic EP encoder is cleanly described, the TopL-BMD algorithm is a real contribution with a reasonable complexity analysis, and the simulation parameters are otherwise concrete (m=1024, CRC=8, L=512). The Monte Carlo construction for the polarized index set is sensible, though it optimizes against a capacity expression that is self-cited from the authors' own [42] rather than derived here.\n\nOther soft spots, in proportion: Lemma 1's posterior probability in Eq. (31) looks wrong as written (it treats the channel as a binary Gaussian channel, but the parity section is a J-user sum; the LLRs in Eq. (21) are for the parity, while Lemma 1 is for the information section—still, the formula does not follow from the preceding definitions). The abstract references a 'Marto Loco' method that never appears in the body. These are fixable, but they support a verdict of 'needs revision.'\n\nBottom line: this paper deserves peer review, not a desk rejection, because the core FFMA-polar idea is plausible and the authors have a track record in this line. But a referee should ask for a corrected simulation with consistent M=KJ (or an explicit explanation of the zero-padded information positions), a derivation or external reference for the capacity, and ideally the simulation code. If the normalization is fixed and the gain survives, it is a useful result for mMTC; if not, the paper still has value as a code construction, just not with the claimed gain.","headline":"Interesting FFMA-polar construction, but the headline 1.25 dB gain rests on an internally inconsistent normalization that must be fixed before the claim is credible.","tokens_in":21471,"tokens_out":12120,"would_cite":false,"duration_ms":128874,"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 polarized element-pair (EP) codes for finite-field multiple-access (FFMA) over a Gaussian multiple-access channel, derives a two-section capacity formula, and reports a 1.25 dB BER gain over polar random spreading for…","keywords":["polarized element-pair code","finite-field multiple access","Gaussian multiple-access channel","channel polarization","successive cancellation list decoding","top L bifurcated minimum distance decoding","power allocation","finite blocklength"],"falsifier":"Reproduce the $J=15$, $K=32$ simulation with the exact parameters reported ($m=1024$, $M=992$, CRC length 8, $L=512$): if TopL-BMD does not reach BER $10^{-5}$ about 1.25 dB below the polar random spreading baseline, the central gain claim fails; a second check is whether a fixed power allocation reproduces the same BER, which would show the optimized power split is not load-bearing.","tokens_in":2077,"feed_emoji":"📡","tokens_out":3250,"duration_ms":108865,"temperature":0.7,"pith_summary":"The paper seeks to establish that the finite-blocklength multiuser coding problem can be solved by a polar-style code built for the Gaussian multiple-access channel rather than for a single-user link. The authors construct polarized element-pair (EP) codes, which treat pairs of field elements as virtual resources that separate users, and place them inside the finite-field multiple-access (FFMA) architecture, where multiuser signals are superimposed before decoding. They derive a capacity formula for the resulting system, design a Monte Carlo construction of the polarized index set and an optimal power split, and propose two decoders: an SCL decoder for larger payloads and a TopL-BMD decoder for small payloads. Simulations for 15 users at 32 bits per user show a BER of $10^{-5}$ at about 1.25 dB lower $E_b/N_0$ than the polar random spreading baseline, which is the concrete claim the paper rises or falls on.","feed_headline":"New multiuser code beats polar spreading by 1.25 dB","feed_subtitle":"For 15 users, the BER 10^-5 point moves 1.25 dB lower in Eb/N0 than polar spreading.","key_machinery":"The load-bearing object is the polarized element-pair (EP) code: the Cartesian product of $M$ element pairs $C_j=(0,\\alpha^{l_{j,1}})$, where each $\\alpha^{l_{j,1}}$ is a row of the $\\kappa$-fold Kronecker matrix $G^{(\\kappa)}$ over $\\mathrm{GF}(2^m)$. A systematic-form generator separates each user's codeword into an information section of length $M=JK$ and a parity section of length $R=m-M$, which lets the transmitter allocate different power to the two sections via $\\mu_{\\mathrm{inf}}$ and $\\mu_{\\mathrm{red}}$. The receiver treats the superimposed signal as a codeword of this EP code; the paper's capacity analysis models the channel as a cascade, and the construction uses a Monte Carlo calculation of polarized subchannel capacities to pick the index set $\\mathcal{A}$ and the power ratio $\\mu_{\\mathrm{pas}}=\\mu_{\\mathrm{inf}}/\\mu_{\\mathrm{red}}$. Two decoders carry the argument: SCL with a path metric, and TopL-BMD, which uses a min-heap to find the $L$ most probable flip sets and then minimum-distance re-encoding.","core_discovery":"The central claim is that a systematic polarized EP code, built from selected rows of the Kronecker generator matrix over $\\mathrm{GF}(2^m)$ and split into an information section and a parity section, makes FFMA operate effectively over the GMAC at finite blocklength. The paper decomposes the GMAC into a binary-input approximate-symmetric channel followed by a multiple-input non-symmetric channel, and expresses the total capacity as $C_{\\mathrm{tot}}=\\frac{JK}{2}\\log_2(1+\\mu_{\\mathrm{inf}}P_{\\mathrm{avg}}/\\sigma^2)+\\frac{R}{2}\\log_2(1+J\\mu_{\\mathrm{red}}P_{\\mathrm{avg}}/\\sigma^2)$, with $\\mu_{\\mathrm{inf}}$ and $\\mu_{\\mathrm{red}}$ as polarization-adjusted power factors obeying $mP_{\\mathrm{avg}}=K\\mu_{\\mathrm{inf}}P_{\\mathrm{avg}}+R\\mu_{\\mathrm{red}}P_{\\mathrm{avg}}$. With that construction, the paper reports that the SCL decoder reaches BER $10^{-5}$ about 1.25 dB lower in $E_b/N_0$ than polar random spreading for 15 users and $K=32$, and that the TopL-BMD decoder gives comparable performance for small payloads.","pith_inferences":["The capacity formula implies per-user capacity grows only logarithmically in $J$, so the reported 1.25 dB gain is not predicted to persist at much larger user counts; a natural test is to run the same comparison at $J=50$ or $J=100$.","The Monte Carlo construction could be swapped for a deterministic partial-order construction to test whether the index set, rather than the FFMA structure, is what produces the reported gain.","The same EP-code machinery could be tried with LDPC or BCH inner codes to isolate the polarization contribution from the finite-field multiplexing contribution."],"forward_implications":["For $K=64$ and SCL decoding, the PA-FFMA system beats polar random spreading by about 1.5 dB at $J=5$, 2.5 dB at $J=10$, and keeps functioning at $J=15$ where the baseline fails to reach BER $10^{-5}$.","For $K=32$, TopL-BMD slightly outperforms SCL at small user counts and delivers the headline 1.25 dB gain at $J=15$ over polar random spreading.","The SCL decoder's complexity is $O(JR + Lm\\log m)$, independent of the number of users except in the LLR calculation, and is lower than the iterative SIC-based polar spreading baseline.","The optimal power allocation shifts more power to the parity section as $E_b/N_0$ or the user count grows, because the parity-section capacity then dominates the information-section capacity.","The system still functions at $J=31$ users with $K=32$, where the polar random spreading baseline fails to reach a BER of $10^{-5}$."],"supporting_citations":[{"why":"Defines the symbol-wise FFMA framework and the BMD decoding that the polarized EP code extends.","marker":"[40]"},{"why":"Defines codeword-wise FFMA and the Monte Carlo power allocation used here.","marker":"[41]"},{"why":"Supplies the FFMA capacity decomposition into a binary-input channel and a multiple-input channel that Eq. (28) builds on.","marker":"[42]"},{"why":"Provides the polar random spreading baseline that the BER comparisons must beat.","marker":"[15]"},{"why":"Introduces polar codes and channel polarization, the foundation of the Kronecker construction.","marker":"[21]"},{"why":"Gives the transformation to systematic polar coding used to split information and parity sections.","marker":"[44]"},{"why":"Provides list decoding of polar codes underlying the SCL decoder.","marker":"[22]"},{"why":"Provides the polarized-channel capacity formula used in the Monte Carlo construction.","marker":"[48]"},{"why":"Introduces CRC-aided polar decoding, extended here to the CRC-aided systematic EP code.","marker":"[23]"},{"why":"Underlies the min-heap subset search used by the TopL algorithm to generate candidate flip sets.","marker":"[50]"}],"fun_headline_variants":["Polarized EP code gives 1.25 dB gain in multiuser GMAC","New FFMA code beats polar spreading by 1.25 dB for 15 users","Element-pair codes for Gaussian MAC outperform polar spreading","1.25 dB coding gain with polarized EP codes in FFMA","Multiuser EP codes gain 1.25 dB over polar random spreading"],"cache_read_input_tokens":23552,"weakest_assumption_plain":"The code construction and power split are chosen using idealized successive-cancellation analysis and a capacity proxy, and the results assume those choices remain near-optimal for the practical SCL and TopL-BMD decoders at the finite blocklengths simulated.","fun_headline_variants_meta":{"raw":{"variants":["Polarized EP code gives 1.25 dB gain in multiuser GMAC","New FFMA code beats polar spreading by 1.25 dB for 15 users","Element-pair codes for Gaussian MAC outperform polar spreading","1.25 dB coding gain with polarized EP codes in FFMA","Multiuser EP codes gain 1.25 dB over polar random spreading"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000237,"raw_usage":{"total_tokens":1567,"prompt_tokens":1064,"completion_tokens":503,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":680,"completion_tokens_details":{"reasoning_tokens":404}},"tokens_in":680,"tokens_out":503,"duration_ms":4360,"temperature":1.0,"reasoning_tokens":404,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:49:59.429832+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Reproduce the $J=15$, $K=32$ simulation with the exact parameters reported ($m=1024$, $M=992$, CRC length 8, $L=512$): if TopL-BMD does not reach BER $10^{-5}$ about 1.25 dB below the polar random spreading baseline, the central gain claim fails; a second check is whether a fixed power allocation reproduces the same BER, which would show the optimized power split is not load-bearing.","supporting_citations":[{"cited_title":"Polar coding and random spreading for unsourced multiple access,","cited_arxiv_id":null,"evidence_quote":"Provides the polar random spreading baseline that the BER comparisons must beat."},{"cited_title":"Systematic polar coding,","cited_arxiv_id":null,"evidence_quote":"Gives the transformation to systematic polar coding used to split information and parity sections."},{"cited_title":"List decoding of polar codes,","cited_arxiv_id":null,"evidence_quote":"Provides list decoding of polar codes underlying the SCL decoder."},{"cited_title":"How to construct polar codes,","cited_arxiv_id":null,"evidence_quote":"Provides the polarized-channel capacity formula used in the Monte Carlo construction."},{"cited_title":"Crc-aided decoding of polar codes,","cited_arxiv_id":null,"evidence_quote":"Introduces CRC-aided polar decoding, extended here to the CRC-aided systematic EP code."}],"review_version":1}