{"id":"62f80456-d287-47a1-9ac8-1e9787aafd4e","arxiv_id":"2505.06867","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Extended deep polar codes concatenate the intermediate polar codewords of each pre-transform layer to achieve non-power-of-two blocklengths with a soft-output SCL decoder, outperforming conventional rate-matching at medium to high rates.","lead":"This paper proposes a rate-matching method for deep polar codes, enabling codewords of arbitrary length by concatenating the intermediate codewords generated at each layer of the deep polar encoder. The method shows coding gains over standard repetition, puncturing, and shortening at medium to high code rates, especially when the desired length is just above a power of two.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed BLER gains may partly reflect the comparison protocol: puncturing and shortening are run with half the list size and a double-length mother code, so the headline improvement over them is not yet cleanly separated from the list-size/mother-length trade.","rationale":"The reader's weakest-assumption pick, the Gaussian approximation for the soft-output messages, is a real theoretical weakness and is acknowledged in the paper itself. However, the central claim is empirical: the proposed extended deep polar code achieves substantially lower BLER than repetition, puncturing, and shortening at the same overall rate. That claim is supported by the simulations in Figs. 7-11, not by the DEGA analysis. Even if the Gaussian approximation biases the optimized Kq values, the simulated BLER curves still describe the actual performance of the constructed codes; an imperfect design rule would not by itself overturn the measured gains. The more load-bearing unverified condition is the fairness of the simulation protocol. In Section V and Figs. 8-11, puncturing and shortening are always run with half the list size of the proposed scheme and with a doubled mother code length. This is a complexity-normalization choice, but for short blocklength SCL decoding the equalization is only approximate, and list-size differences of 1 versus 2 or 4 versus 8 can materially change BLER at operating points near 10^-3. If the baselines were given the same list size, their BLER would improve and the claimed margin could shrink. This is a concrete, testable concern rather than an internal inconsistency, so the appropriate verdict remains CONDITIONAL pending an equal-list-size check. I partially agree with the reader: the list-size issue was noted in their rationale, but it was not selected as the weakest assumption.","tokens_in":32326,"tokens_out":16096,"duration_ms":190829,"concrete_test":"Re-run the Fig. 10 comparison for M=272 and M=528 (and preferably all subfigures) with puncturing and shortening using the same list size L as the proposed scheme, i.e., L=2 and L=8, keeping the mother length 2*N0, and recompute the required SNR at BLER 10^-3. Also run one variant in which the puncturing/shortening rate profiles are optimized rather than fixed to the 5G quasi-uniform choice. If the proposed gain over puncturing and shortening shrinks below roughly 0.2-0.3 dB or reverses, the headline comparison is not established as stated; if the margin persists under equal list size, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central empirical claim rests on Figs. 8-11, where the proposed extension and repetition use mother length N0 and list size L, while puncturing and shortening use mother length 2*N0 and list size L/2. For example, Fig. 8 (M=152) uses N0=128 with L=2 for the proposed scheme versus N=256 with L=1 for puncturing/shortening; Fig. 10 (M=272) uses N0=256 with L=2 or L=8 versus N=512 with L=1 or L=4. The authors justify this by asymptotic complexity O(L*N*logN), but at short blocklengths this equalization is not exact: SCL list-management and path-sorting overhead grows more than linearly in L, punctured/shortened decoders can exploit known or punctured positions to reduce effective work, and L=1 removes essentially all list diversity from the baselines while the proposed scheme is given L=2 or L=8. Since the headline claim is lower BLER at the same overall rate, the load-bearing assumption is that the comparison isolates the coding/extension gain rather than an unequal decoder budget. The DEGA Gaussian approximation of Section III-C affects the design rule and the predicted error probabilities, but it does not directly threaten the measured BLER comparisons; a suboptimal design choice would weaken the design algorithm, not the empirical existence of a coding gain. Thus the comparison-fairness issue is the more immediate threat to the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a rate-matching method for deep polar codes when the desired codeword length M is slightly larger than a power of two. The construction concatenates the layer outputs of a deep polar code, c = [c0, c1, ..., cQ], where each cq is a polar codeword generated by a pre-transform layer; the total length is M = N0 + N1 + ... + NQ. The authors develop a soft-output successive cancellation list (SoSCL) decoder that obtains soft information from the extension layers and feeds it into a modified SCL decoder for the main layer. They give a density-evolution under Gaussian approximation (DEGA) analysis of the decoding error probability, use it to design the information and connection index sets, and propose both an exhaustive and a greedy rate-profiling algorithm for multi-layer configurations. Simulations compare the proposed scheme with repetition, puncturing, and shortening over BI-AWGN channels and report BLER gains, especially at medium to high code rates.","tokens_in":32604,"tokens_out":8892,"duration_ms":96755,"significance":"If the empirical gains hold under a fair comparison, the paper offers a useful low-complexity rate-matching option for short-blocklength codes, particularly when the target length is just above a power of two. The construction is simple, the decoding overhead over the component polar decoders is modest, and the proposed greedy algorithm makes multi-layer design tractable. The paper also provides a concrete error-probability design framework, which is a strength even if the Gaussian approximation is heuristic. The main significance is therefore practical: a potential alternative to puncturing and shortening for short-blocklength URLLC-style codes. The novelty relative to the authors' prior work on deep polar codes is incremental but sufficient for a rate-matching contribution.","major_comments":[{"comment":"The headline comparison is not complexity-fair at short blocklengths. Puncturing and shortening are run with mother code length 2*N0 and list size L/2, while the proposed scheme and repetition use mother code length N0 and list size L. The paper justifies this by the asymptotic complexity O(L*N*log N), but at the simulated lengths the decoder budgets are not equal: SCL path management and sorting grow more than linearly in L, and the L=1 baselines have essentially no list diversity, while punctured/shortened decoders can also exploit known or punctured positions to reduce effective work. Since the central claim is a BLER gain at the same overall rate, the comparison should either use the same list size for all schemes or a carefully complexity-matched (energy/latency) accounting. Without this, part of the reported gain over puncturing and shortening may be an artifact of the unequal decoder budget rather than a coding gain of the proposed extension.","section":"Section V, Figs. 8-11"},{"comment":"There is an internal inconsistency about which soft information is used. Equation (23) combines the SoSCL output only at connection positions A1,I1, i.e., positions corresponding to information bits of layer 1, and sets Lambda_i = 0 elsewhere. However, Remark 1 states that the soft information vector encodes partial reliability knowledge 'including for subsequent frozen bits indexed by A1,F1' and that this is precisely what improves decoding of the first connection bit. Since c1 = u1*G1^T and frozen bits of u1 still affect many bits of c1, the soft outputs at A1,F1 are not known a priori and appear to carry useful information. The authors should clarify which mechanism is actually implemented; if only A1,I1 is used, the decoder is not exploiting the full soft-output structure, and the claimed benefit in Remark 1 is not realized. The same issue applies to Eq. (39) in the multi-layer case.","section":"Section III-B, Eq. (23) and Remark 1"},{"comment":"The DEGA analysis assumes that the right-to-left messages satisfy Rd,i ~ N(eta_d,i, 2*eta_d,i), while the text acknowledges that Rn,i is a binary-valued quantity and hence not Gaussian. The design metric (30) and all subsequent rate-profiling in Section III-D rely on the resulting eta values. No validation of this approximation is provided, for example by comparing predicted eta_0,i with empirical means from SoSCL decoding, or by checking whether DEGA-optimized Kq values coincide with parameters found by minimizing the simulated BLER. Because the rate-profiling framework is a claimed contribution, this gap is load-bearing for the design algorithm, even though the measured BLER comparisons themselves do not depend on the approximation.","section":"Section III-C, Eqs. (28)-(30)"},{"comment":"The index N1-i in eta_0,N1-i is out of range for i=0. For a vector of length N1, the reversal mapping should presumably be N1-1-i. Please correct the formula and verify the index arithmetic, since Eq. (30) is a central design equation and the out-of-range index affects the stated product over i in I1.","section":"Eq. (30)"}],"minor_comments":[{"comment":"The greedy stopping rule 'the largest Kq such that Pe,q < Pe,q+1' is justified by empirical observations rather than derived. Since the algorithm is a heuristic, please state explicitly what guarantees it does and does not provide, and ideally add a sensitivity check against the exhaustive search for a small case.","section":"Algorithm 2, Section IV-D2"},{"comment":"The BEC example would be easier to follow if the bit indices were defined explicitly; the text refers to 'u3' and 'u4' without stating the index ordering used to obtain the erasure probabilities {0.06, 0.44, 0.56, 0.94} for N1=4.","section":"Section III-E, Example"},{"comment":"There are typographical errors that should be corrected: 'adoptation' in the introduction, 'detph' in the caption of Fig. 1, and 'blocklenghts' in the conclusion.","section":"Throughout"},{"comment":"The caption and text say Fig. 7 uses SC decoding with an 11-bit CRC, but the legend includes a curve labeled 'Approximated by Eq. (30)'. It would be helpful to state explicitly in the caption that the approximation treats CRC bits as information bits, as the text does.","section":"Section V, Fig. 7"}],"recommendation":"major_revision","confidential_remarks":"The comparison-fairness issue in Section V is the most serious concern because it directly affects the paper's main empirical claim. The DEGA approximation issue is secondary but should be addressed because the design framework is a stated contribution. The relationship to the authors' previous deep polar codes [14,15] is incremental; the rate-matching aspect is the new contribution and should be evaluated on that basis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the construction is real and the paper is worth refereeing. The idea is to use the intermediate codewords of a deep polar encoder as extension bits, concatenate them with the main codeword, and feed soft-output SCL information from the extension layers back into the main decoder. That is new relative to the cited extension literature, which mostly targets conventional polar codes. The simulations support the main claim: at medium and high rates, the extended code beats repetition at the same mother length and same list size. That comparison is clean, and it is enough to establish the value of the method.\n\nThe gains over puncturing and shortening are also shown, but those baselines get a longer mother code and half the list size, with complexity equalized only asymptotically. At short blocklengths, list management overhead and effective decoder work do not scale linearly, so L=1 versus L=2 is not just a 2x complexity difference. I would not treat the puncturing/shortening gap as precisely quantified. I also would not treat it as fatal, because the repetition comparison is fair and already demonstrates coding gain.\n\nThe DEGA analysis is honestly flagged as optimistic: the authors assume Gaussian messages for the soft-output R values after noting that Rn,i is actually binary, and they explicitly say the resulting estimate may overestimate reliability. That weakens the design metric, not the measured BLER. The greedy algorithm is justified by empirical observations rather than derived; that is a real limitation, but a minor one for a practical design paper.\n\nWriting is a bit loose in places (\"adoptation,\" \"blocklenghts\"), but the encoding, decoding, and simulations are internally consistent. The citation pattern is appropriate; the self-citations point to the authors' own deep polar code papers, which is exactly what this work builds on.\n\nFor the right reader, this is useful: anyone working on rate-matching for short-blocklength codes, especially URLLC, will want to know about this construction. It deserves a serious referee. I would ask the authors to rerun or supplement the puncturing/shortening comparisons with equal list sizes or a more careful complexity accounting, and to add a sensitivity check on the Gaussian approximation. Neither issue undermines the core idea.","headline":"A genuinely new extension scheme for deep polar codes with a clean gain over repetition and a messier, but not fatal, comparison against puncturing and shortening.","tokens_in":33136,"tokens_out":2138,"would_cite":true,"duration_ms":24514,"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":"The paper claims that concatenating the layer outputs of a deep polar encoder yields a rate-matched code that outperforms repetition, puncturing, and shortening, especially at medium to high code rates.","keywords":["deep polar codes","rate matching","code extension","successive cancellation list decoding","soft-output decoding","density evolution","Gaussian approximation","short blocklength codes"],"falsifier":"Run Monte Carlo decoding of a density-evolution-optimized extended deep polar code at high SNR, compare measured BLER with the paper's bound, and instrument the SoSCL decoder to record the empirical mean and variance of each right-to-left message. If the variance is not close to twice the mean, or if a grid search over the split $K_1$ finds a rate profile with lower BLER than the optimized one, the Gaussian approximation is the weak link.","tokens_in":32101,"feed_emoji":"📡","tokens_out":6508,"duration_ms":62831,"temperature":0.7,"pith_summary":"The paper offers a way to give deep polar codes—codes built by stacking polar transformation layers before the final polar transform—arbitrary blocklengths without falling back on the usual rate-matching tricks. When the target length $M$ is only a little larger than a power of two, it forms the extended code by concatenating the codewords produced by each layer of the deep polar encoder. Because those layer codewords are themselves polar codewords, they can be decoded with soft-output successive cancellation list decoding, and their reliability values are fed into the main decoder as extra information. The paper argues and simulates that this lowers block error rate relative to repetition, puncturing, and shortening, especially as the code rate climbs, while avoiding the larger mother code that puncturing and shortening require.","feed_headline":"Extended deep polar codes beat repetition, puncturing, shortening","feed_subtitle":"For lengths just above a power of two, soft information from extension layers lifts reliability without decoding a larger mother code.","key_machinery":"The carrying object is the concatenation of layer outputs: each extension codeword $c_q$ is a polar codeword produced by a transposed polar transform $G_q^\\top$, and the same bits are embedded in layer 0's input at connection positions $A_q$. The decoding mechanism is soft-output SCL (SoSCL), which returns reliability values (LLRs) for the connection bits and adds them to the main decoder's LLRs before path-metric updates. The design mechanism is density evolution under Gaussian approximation (DEGA): the recursion that propagates the mean $\\eta$ of the right-to-left messages, together with the conservative block error rate bound used to choose which indices are information, connection, or frozen.","core_discovery":"The central claim is that the layered structure of a deep polar code can itself be the rate-matching resource. Treating the output of each pre-transform layer as an extra polar codeword, the paper concatenates $c_0$ (the main layer output) with $c_1,\\ldots,c_Q$, so the transmitted word is $c=[c_0,c_1,\\ldots,c_Q]$ and the length $M$ is the sum of the layer lengths, no longer restricted to a power of two. The extended pieces are decoded by soft-output successive cancellation list decoding to produce soft LLRs for the connection bits they share with layer 0; these LLRs are added to the main decoder's bit LLRs. With information and connection sets chosen by a density-evolution-based error-probability analysis, the resulting codes have lower block error rate than repetition, puncturing, and shortening in medium-to-high rate regimes, and a greedy version of the design search makes multi-layer extension practical.","pith_inferences":["Beyond the paper: the same concatenate-the-layer-outputs idea should transfer to other pre-transformed polar constructions whose intermediate stages produce genuine polar codewords; the essential condition is only that the extension pieces are decodable polar codewords whose soft outputs touch the main codeword's input bits.","Beyond the paper: the paper's own optimistic versus conservative error expressions suggest a hybrid design rule—use the optimistic estimate only for layers whose genie-aided soft output is likely reliable, and the conservative bound elsewhere—that could improve the rate profile further.","Beyond the paper: if soft-output decoding of a very small extension layer (for example $N_q = 1$ or $2$) is unreliable, plain LLR combining or treating that layer as repetition is likely better; the paper mentions this direction and leaves it open, so a direct comparison at $N_q=1$ would be a natural next experiment."],"forward_implications":["Setting $M = N_0 + N_1$, the decoder works at the target length but only decodes the $N_0$-sized main block plus soft outputs from the $N_1$-sized extension, so the complexity overhead over a length-$N_0$ polar decoder is small.","The advantage over repetition, puncturing, and shortening grows with code rate, because at high rates puncturing and shortening delete a large fraction of the mother codeword while the extension pieces still deliver useful reliability information.","Multi-layer extension works for lengths such as $M=280=256+16+8$ and $M=304=256+32+16$, and the greedy rate-profiling algorithm matches near-exhaustive search while cutting design complexity from exponential to linear in the number of layers.","The method plugs into the 5G NR reliability sequence and CRC-aided SCL decoding, so it can reuse existing polar-code infrastructure."],"supporting_citations":[{"why":"Defines deep polar codes and their multi-layer polar kernel transformation, the structure being extended here.","marker":"[14]"},{"why":"Introduces the sparsely pre-transformed polar-code variant whose transposed pre-transform is used in the extension layers.","marker":"[15]"},{"why":"Supplies the soft-output SCL decoding algorithm that produces the LLRs fed into the main decoder.","marker":"[27]"},{"why":"Provides the density-evolution method used to estimate LLR means in the error analysis.","marker":"[29]"},{"why":"Gives the Gaussian-approximation construction and block error rate analysis that the paper modifies for soft information.","marker":"[30]"},{"why":"Defines the 5G NR reliability sequence, CRC polynomial, and rate-matching procedures used for simulation and comparison.","marker":"[2]"},{"why":"Proposes structural extension of polar codes via simplex kernels, the prior extension approach this paper is contrasted with.","marker":"[26]"},{"why":"Derives the LLR-based path metric used to combine the soft information into SCL decoding.","marker":"[33]"}],"fun_headline_variants":["Rate-match deep polar codes by concatenating layer outputs","Soft-output SC list decoding unlocks extended deep polar codes","Deep polar extension beats repetition and shortening at high rates","Layered concatenation gives arbitrary lengths to deep polar codes","For lengths just above a power of two, extend deep polar codes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analysis assumes that every internal soft message in the decoder is Gaussian with variance exactly twice its mean, even though the messages at the decoder's edge are binary-valued known quantities; if that approximation fails, the predicted error rates and the chosen rate splits could be off.","fun_headline_variants_meta":{"raw":{"variants":["Rate-match deep polar codes by concatenating layer outputs","Soft-output SC list decoding unlocks extended deep polar codes","Deep polar extension beats repetition and shortening at high rates","Layered concatenation gives arbitrary lengths to deep polar codes","For lengths just above a power of two, extend deep polar codes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000554,"raw_usage":{"total_tokens":2616,"prompt_tokens":901,"completion_tokens":1715,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":517,"completion_tokens_details":{"reasoning_tokens":1635}},"tokens_in":517,"tokens_out":1715,"duration_ms":13842,"temperature":1.0,"reasoning_tokens":1635,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:31:10.874965+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Monte Carlo decoding of a density-evolution-optimized extended deep polar code at high SNR, compare measured BLER with the paper's bound, and instrument the SoSCL decoder to record the empirical mean and variance of each right-to-left message. If the variance is not close to twice the mean, or if a grid search over the split $K_1$ finds a rate profile with lower BLER than the optimized one, the Gaussian approximation is the weak link.","supporting_citations":[{"cited_title":"Deep polar codes,","cited_arxiv_id":null,"evidence_quote":"Defines deep polar codes and their multi-layer polar kernel transformation, the structure being extended here."},{"cited_title":"Sparsely pre-transformed polar codes for low-latency SCL de- coding,","cited_arxiv_id":null,"evidence_quote":"Introduces the sparsely pre-transformed polar-code variant whose transposed pre-transform is used in the extension layers."},{"cited_title":"Soft-out successive cancelation decoding and its list version for polar codes,","cited_arxiv_id":null,"evidence_quote":"Supplies the soft-output SCL decoding algorithm that produces the LLRs fed into the main decoder."},{"cited_title":"Construction and block error rate analysis of polar codes over AWGN channel based on gaussian approximation,","cited_arxiv_id":null,"evidence_quote":"Gives the Gaussian-approximation construction and block error rate analysis that the paper modifies for soft information."},{"cited_title":"NR; multiplexing and channel coding,","cited_arxiv_id":null,"evidence_quote":"Defines the 5G NR reliability sequence, CRC polynomial, and rate-matching procedures used for simulation and comparison."},{"cited_title":"Structural extension of polar codes via simplex kernels,","cited_arxiv_id":null,"evidence_quote":"Proposes structural extension of polar codes via simplex kernels, the prior extension approach this paper is contrasted with."},{"cited_title":"LLR-based successive cancellation list decoding of polar codes,","cited_arxiv_id":null,"evidence_quote":"Derives the LLR-based path metric used to combine the soft information into SCL decoding."}],"review_version":1}