{"id":"167c81f8-c598-4b5a-9f55-fbc2d93d9f9c","arxiv_id":"2501.04921","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"New families of concatenated entanglement-assisted quantum codes are claimed to beat known code tables, and the paper attempts to show such codes asymptotically reach the entanglement-assisted Gilbert-Varshamov bound.","lead":"This paper builds new quantum error-correcting codes by stacking two entanglement-assisted codes, using less rigid outer codes to obtain codes with larger minimum distances than previously tabulated codes. If the constructions hold, the result is a richer toolbox of quantum codes for communication and fault-tolerant computation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Asymptotic GV-bound proof in Section V rests on a false exponent identity in Lemma 13; the claimed meeting of the EA-GV bound is not established.","rationale":"The Reader's verdict identifies two defects: (i) the hull-dimension/existence gap in Lemmas 8-9 and the tables, and (ii) a false algebraic equality in Lemma 13. Both are legitimate, but the single most load-bearing issue for the paper's central asymptotic claim is the Lemma 13 error. The finite-length tables could in principle be salvaged by citing or constructing AMDS codes with the needed hulls, and several individual constructions are plausibly correct; but the GV-bound theorem is a flagship claim stated in the abstract and Section V, and its proof collapses at a concrete algebraic step. The false identity is not a cosmetic typo: it changes the exponent's dependence on rates and entanglement parameters, so the resulting union-bound estimate and the conclusion P -> 0 are not justified. I therefore agree with the Reader's REJECT verdict, with the emphasis placed on the asymptotic proof rather than on the hull-existence point. Note also that some details in the paper are healthy: the concatenation framework (Lemma 3), the maximal-entanglement preservation result (Lemma 10), and the explicit families in Proposition 1 follow from cited methods, so the paper is not without useful content. But the central claim of asymptotically attaining the EA-GV bound is unsupported, and the stated parameter improvements depend on an existence assumption that is not proven for all displayed hull dimensions.","tokens_in":24554,"tokens_out":1706,"duration_ms":14530,"concrete_test":"Independently recompute the exponent identity in Lemma 13 for a generic instance, e.g., take k1=2, n1=4, c1=0, and k2=1, n2=23, c2=0, which matches the first row of Table I for the unassisted outer case. Verify whether 4^{-(bar_k1 r2 + r1 n2)} equals 2^{-(re + ce)} under the definitions re = n1*n2 - bar_k1*bar_k2 and ce = c1*n2 + c2*bar_k1. If the two expressions differ, the proof of Eq. (36) is invalid and Theorem 1 is unsupported unless a corrected derivation is supplied.","verdict_should_be":"REJECT","load_bearing_attack":"The central asymptotic claim is that random EACQCs attain the quantum GV bound for EAQECCs (Section V, Theorem 1). The proof depends on Lemma 13, where the final displayed identity is 4^{-(bar_k1 r2 + r1 n2)} = 2^{-(re + ce)}. This equality is false: re = ne - bar_k1*bar_k2 and ce = c1*n2 + c2*bar_k1, while bar_k2 = 2*k2 - n2 + c2, so re + ce = n1*n2 - bar_k1*bar_k2 + c1*n2 + c2*bar_k1. Expressing this in terms of r1 = n1 - k1 and r2 = n2 - k2 gives a different expression; for example, when c1 = c2 = 0 the claimed equality would be 4^{-(k1*r2 + r1*n2)} = 2^{-(n1*r2 + r1*n2)}, which holds only if k1 = n1/2, i.e., inner rate 1/2. Thus the bound in Eq. (36) is not derived from the preceding line, and the subsequent entropy estimates in Lemma 14 and the vanishing probability in Theorem 1 do not follow. Because the GV-bound theorem is the paper's headline asymptotic result, this is a load-bearing defect. It is a specific algebraic error, not a disagreement with consensus; the claimed theorem may be true, but this manuscript does not prove it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes constructions of entanglement-assisted concatenated quantum codes (EACQCs) using almost-MDS, ℏ-MDS, and algebraic-geometry codes as the outer component codes, building on the EACQC framework of the authors' earlier work [23]. It claims new finite-length EACQCs with parameters better than the best known EAQECCs and standard QECCs of the same length and net transmission, gives several maximal-entanglement EACQCs including optimal and almost-optimal examples, constructs two explicit asymptotically good maximal-entanglement families from EAQAG codes, and proves an asymptotic result that EACQCs can attain the entanglement-assisted quantum Gilbert-Varshamov bound.","tokens_in":24828,"tokens_out":33738,"duration_ms":299910,"significance":"The asymptotic GV-bound claim is a substantial potential contribution if fully established, and the explicit asymptotically good maximal-entanglement families in Section IV are a useful addition to the EAQECC literature. I checked the algebra in the asymptotic section in some detail: the exponent identity in Lemma 13, which one might suspect of being false, is in fact correct after expanding \\bar{k}_1=2k_1-n_1+c_1 and \\bar{k}_2=2k_2-n_2+c_2, and the entropy bound in Lemma 14 follows. The maximal-entanglement results in Section IV are also arithmetically consistent. However, the finite-length tables in Section III rely on an unproved freedom in the entanglement parameter c, and this is a load-bearing gap for the paper's central table-based claims.","major_comments":[{"comment":"The constructions in Section III treat the entanglement parameter c as a freely selectable integer in the range 0 ≤ c ≤ n_e − k_e, but the lemmas cited (Lemma 8 and Lemma 9) only establish the existence of some AMDS or genus-2 AG code; they do not control the Hermitian hull dimension c = rank(HH†) of the component code. Since c is determined by the component code, the displayed families such as the example Q2 = [[25, 1+c, 12; c]]_4 for 0 ≤ c ≤ 12 are not established for intermediate c. The proof must either provide a construction or citation for AMDS and ℏ-MDS codes with every prescribed Hermitian hull dimension, or the tables must be restricted to c values for which existence is already known (e.g., c = n_e − k_e via the LCD-code equivalence in [21] or c = 0 via dual-containing codes).","section":"Section III, Lemma 8 and Lemma 9; Tables I-III"},{"comment":"The asymptotic proof does not specify the growth regime of n1 and n2 in the step where the factor [1 + 4^{r1}(1−δ_e)^{n1}]^{n2} is bounded by e^c. The text 'c = τ n1 n2 is a constant' appears to be a typographical loss of the intended τ^{n_1} n_2, and keeping this product constant while n_e = n1 n2 → ∞ forces n1 ∼ log n2. The proof should state this growth explicitly and verify that the rate R_e − C_e converges to the claimed limit under such a choice; as written, the existence of a sequence satisfying both conditions is not demonstrated.","section":"Section V, Theorem 1"}],"minor_comments":[{"comment":"Several superscripts and subscripts are missing or garbled in the text, for example '4¯k1' for 4^{\\bar{k}_1} and 'τ n1 n2' for τ^{n_1} n_2; a careful typesetting pass is needed.","section":"Throughout"},{"comment":"The operations called 'Code Extension' and 'Code Expurgation' are used throughout the tables but are never defined; please explain how they act on the component codes and why they preserve the stated distance and entanglement parameters.","section":"Tables I-III"},{"comment":"The sentence 'If ui = 0 for some nonzero ui' is garbled and should be rewritten to distinguish the information part and the parity part of the vector u_i when computing the inner-code syndrome probability.","section":"Section V"},{"comment":"The symbol ℏe is used both for the EA quantum Singleton defect and in the term 'ℏe-EAQMDS', which is confusing; please use separate notations for the defect and for the code family.","section":"Section III, Eq. (8)"},{"comment":"The abstract contains a sentence fragment beginning 'Because the range of code length ...'; please rephrase to form a complete sentence.","section":"Abstract"},{"comment":"The statement about bold numbers in brackets does not match the plain-text table; please indicate clearly how the comparison with optimal classical quaternary codes in [37] is performed.","section":"Table IV"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a sound asymptotic core; the reported allegation that Lemma 13 contains a false exponent identity appears to be based on a misreading, since the identity is correct after substitution. The main obstacle is the hull-dimension freedom in Section III, which undermines the table-based finite-length claims. If the authors can either prove prescribed-hull AMDS/AG codes or substantially restrict the claimed c ranges, the paper may become publishable. The asymptotic GV proof also needs a small but important clarification of the n1, n2 growth regime."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the genuinely new material is the use of AMDS, h-MDS, and AG codes as outer codes for EACQCs, plus the observation that maximal entanglement survives concatenation. Second, neither headline claim is established: the finite-length tables overclaim because the entanglement parameter c is treated as freely adjustable without a construction for the hull dimension, and the asymptotic GV theorem rests on a false algebraic identity.\n\nThe clean part is Section IV. Lemma 10 says that if both component codes have maximal entanglement, so does the EACQC — that's a short, correct step. The resulting optimal codes such as [[26,2,20;24]], [[36,2,28;34]], and [[46,2,36;34]] are specific and checkable. The asymptotically good maximal-entanglement families from AG outer codes also look right; they follow from known EAQAG bounds and the same Lemma 10.\n\nThe soft spots are proportional and they are in the two places the authors advertise most. Section III's tables need AMDS and h-MDS component codes with every prescribed hull dimension c. Lemma 8 and Lemma 9 only prove existence for some c. The examples and tables vary c freely from 0 up to ne−ke. Without a construction or a citation for hull dimensions in that range, the claimed improvements over the best known codes are not supported. That is not a nitpick; it is the load-bearing part of Section III.\n\nSection V has a sharper problem. Lemma 13 ends with the identity 4^{-(bar_k1 r2 + r1 n2)} = 2^{-(re+ce)}. This equality is false. In the c1=c2=0 case it becomes 4^{-(k1 r2 + r1 n2)} = 2^{-(n1 r2 + r1 n2)}, which only holds when k1 = n1/2. So Eq. (36) does not follow from the preceding line, and the subsequent entropy estimates in Lemma 14 and Theorem 1 do not go through. The theorem may be true, but this manuscript does not prove it.\n\nI agree with your reader's assessment on both counts. The paper is not circular — it builds on known AMDS and AG results and does not fit constants — but the two central claims are unsupported as written.\n\nWho should read this? Someone working on EAQECCs and concatenated codes might get value from Section IV's constructions and the maximal-entanglement observation. But they should not rely on the tables or the GV result.\n\nRecommendation: this deserves a serious referee rather than a desk reject, because the framework is worth checking and the Section IV results are likely salvageable. My own verdict is reject, with a request to fix the hull-dimension gap and correct Lemma 13 before resubmission.","headline":"The GV-bound proof rests on a false exponent identity, and the finite-length tables rest on an unsupported freedom in the entanglement parameter; Section IV is solid but the headline results need rework.","tokens_in":25366,"tokens_out":2982,"would_cite":false,"duration_ms":28905,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P70","94B27","94B65"],"pacs":["03.67.Pp"],"model":"deepseek-v4-flash","headline":"This paper constructs entanglement-assisted concatenated quantum codes that beat the best previously known quantum codes of the same length and net transmission, and proves such codes can asymptotically attain the quantum…","keywords":["entanglement-assisted concatenated quantum code","almost MDS code","ℏ-MDS code","algebraic geometry code","maximal entanglement","quantum Gilbert-Varshamov bound","concatenated quantum code","EAQECC"],"falsifier":"Compute the Hermitian hull dimension of the specific almost-MDS or ℏ-MDS AG codes used in the tables (e.g., the [[25,1,12;12]]$_{4}$ outer code) and check whether the resulting $c$ matches the claimed value; a mismatch would change the code parameters. Alternatively, consult an updated online code table for a length such as $N=100$ with net transmission $2$ and see whether any listed EAQECC or QECC already has minimum distance at least $24$, which would invalidate the claimed improvement.","tokens_in":24310,"feed_emoji":"🔐","tokens_out":3789,"duration_ms":36742,"temperature":0.7,"pith_summary":"The paper builds quantum error-correcting codes by concatenating two entanglement-assisted quantum codes (EACQCs). Using almost-MDS and ℏ-MDS classical codes as outer components gives much more freedom in code length than the MDS codes used before, and yields explicit families whose minimum distance exceeds the best known EAQECCs and standard QECCs at the same length and net transmission. The paper also shows that maximal entanglement propagates through concatenation, giving several optimal or near-optimal maximal-entanglement codes, and it proves that suitably randomized EACQCs asymptotically meet the quantum Gilbert-Varshamov bound for EAQECCs.","feed_headline":"Two-level entanglement codes outdo all known quantum codes","feed_subtitle":"New families also reach the quantum Gilbert-Varshamov limit asymptotically while using maximal entanglement.","key_machinery":"The central mechanism is the EACQC concatenation rule $Q_e = [[n_1 n_2, k_1 k_2, d_e \\ge d_1 d_2; c_e]]_q$ with $c_e = c_1 n_2 + c_2 k_1$, which converts an inner EAQECC and an outer EAQECC into a longer code whose entanglement cost is a weighted sum. The outer codes are chosen from almost-MDS and ℏ-MDS codes, whose lengths can reach $q^2 + 2q + 1$ and $q^2 + 4q + 1$ respectively (via algebraic-geometry code bounds), much longer than the MDS-conjecture limit. Maximal entanglement is propagated by Lemma 10, and the asymptotic GV-bound result follows from a random ensemble counting argument using the generating function $\\Psi_t(x) = \\binom{n_2}{t}[(1+3x)^{n_1} - 1]^t$ to bound the average number of low-weight errors.","core_discovery":"On its own terms, the paper establishes that entanglement-assisted concatenation is a powerful construction tool: by choosing outer codes from almost-MDS, ℏ-MDS, and algebraic-geometry families, it produces EACQCs with parameters better than the previously best known nondegenerate EAQECCs and standard QECCs of the same length and net transmission. It further shows that if both component codes consume maximal entanglement, then the concatenated code does too, and it uses this to construct three new optimal maximal-entanglement EACQCs and several codes whose distance is one less than optimal. Finally, it proves that EACQCs can attain the quantum Gilbert-Varshamov bound for EAQECCs asymptotically, extending a classical result of Blokh and Zyablov to the entanglement-assisted setting.","pith_inferences":["If constructions of almost-MDS or ℏ-MDS codes with every prescribed Hermitian hull dimension become available, the table improvements in this paper would likely extend to many more lengths and entanglement parameters.","The random ensemble proof ignores the internal weight structure of subblocks and may underestimate the frequency of degenerate errors, suggesting degenerate EACQCs could potentially beat the nondegenerate EAQECC bounds beyond the GV point.","The length-flexibility advantage of AMDS outer codes could be combined with non-binary inner codes to generate higher-rate families, possibly improving the asymptotic rate region for maximal-entanglement codes.","The claimed improvements depend on the online code tables being current as of 2025; future updates of those tables may shrink or eliminate some of the listed distance gaps."],"forward_implications":["New EACQCs with larger minimum distance than the best known EAQECCs and QECCs of the same length and net transmission are now available for several lengths around 90 to 160.","Maximal-entanglement EACQCs inherit maximal entanglement from their components, enabling constructions that achieve the entanglement-assisted quantum capacity for suitable channels.","Two explicit infinite families of asymptotically good binary maximal-entanglement EACQCs with positive net transmission rates exist, with rates lower-bounded by expressions involving $1 - m\\delta - 1/(2^{m/2}-1)$.","The quantum Gilbert-Varshamov bound for EAQECCs is asymptotically attainable by EACQCs, so concatenation does not sacrifice asymptotic performance even when entanglement assistance is priced in.","Code expurgation and extension techniques applied to the new outer codes extend the parameter improvements to a wider range of lengths without changing the net transmission."],"supporting_citations":[{"why":"Establishes the general construction of EAQECCs from any classical linear code without dual-containing constraints.","marker":"[10]"},{"why":"Supplies the optimal entanglement formulas used to compute the required number of ebits.","marker":"[11]"},{"why":"Provides the EACQC concatenation framework and the parameter formula used throughout the paper.","marker":"[23]"},{"why":"Gives the lower bound on the maximum length of almost-MDS codes (Lemma 4) used to justify the long outer codes.","marker":"[29]"},{"why":"Supplies the asymptotic AG-code bound and Riemann-Roch dimension statements used for ℏ-MDS and EAQAG outer codes.","marker":"[35]"},{"why":"Shows that linear codes are equivalent to LCD codes for q>3, enabling maximal-entanglement EAQECC constructions.","marker":"[21]"},{"why":"Provides explicit asymptotically good EAQAG codes with maximal entanglement used as outer codes in the two families.","marker":"[18]"},{"why":"The classical Blokh-Zyablov result on concatenated codes attaining the GV bound, which the paper generalizes to EACQCs.","marker":"[30]"},{"why":"Online bounds tables for best-known minimum distances of linear and quantum codes, used as the baselines for the claimed parameter improvements.","marker":"[37]"}],"fun_headline_variants":["Entanglement codes outdo known quantum codes","Maximal entanglement boosts concatenated codes","EACQCs hit the quantum GV bound asymptotically","Concatenated entanglement codes: optimal parameters","Entanglement concatenation surpasses prior quantum codes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction requires, for each displayed entanglement parameter $c$, the existence of an almost-MDS or ℏ-MDS classical code whose Hermitian hull has dimension exactly $c$, but the paper only proves such codes exist for some values of $c$.","fun_headline_variants_meta":{"raw":{"variants":["Entanglement codes outdo known quantum codes","Maximal entanglement boosts concatenated codes","EACQCs hit the quantum GV bound asymptotically","Concatenated entanglement codes: optimal parameters","Entanglement concatenation surpasses prior quantum codes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001594,"raw_usage":{"total_tokens":6373,"prompt_tokens":986,"completion_tokens":5387,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":5318}},"tokens_in":602,"tokens_out":5387,"duration_ms":43556,"temperature":1.0,"reasoning_tokens":5318,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:25:39.686996+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Hermitian hull dimension of the specific almost-MDS or ℏ-MDS AG codes used in the tables (e.g., the [[25,1,12;12]]$_{4}$ outer code) and check whether the resulting $c$ matches the claimed value; a mismatch would change the code parameters. Alternatively, consult an updated online code table for a length such as $N=100$ with net transmission $2$ and see whether any listed EAQECC or QECC already has minimum distance at least $24$, which would invalidate the claimed improvement.","supporting_citations":[{"cited_title":"Correcting quantum errors with entanglement,","cited_arxiv_id":null,"evidence_quote":"Establishes the general construction of EAQECCs from any classical linear code without dual-containing constraints."},{"cited_title":"Optimal entanglement formulas for entanglement-assisted quantum coding,","cited_arxiv_id":null,"evidence_quote":"Supplies the optimal entanglement formulas used to compute the required number of ebits."},{"cited_title":"Entanglement- assisted concatenated quantum codes,","cited_arxiv_id":null,"evidence_quote":"Provides the EACQC concatenation framework and the parameter formula used throughout the paper."},{"cited_title":"Tsfasman and S","cited_arxiv_id":null,"evidence_quote":"Gives the lower bound on the maximum length of almost-MDS codes (Lemma 4) used to justify the long outer codes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the asymptotic AG-code bound and Riemann-Roch dimension statements used for ℏ-MDS and EAQAG outer codes."},{"cited_title":"Linear codes over Fq are equivalent to LCD codes for q >3,","cited_arxiv_id":null,"evidence_quote":"Shows that linear codes are equivalent to LCD codes for q>3, enabling maximal-entanglement EAQECC constructions."},{"cited_title":"Entanglement-assisted quantum codes from algebraic geometry codes,","cited_arxiv_id":null,"evidence_quote":"Provides explicit asymptotically good EAQAG codes with maximal entanglement used as outer codes in the two families."},{"cited_title":"Existence of linear concatenated binary codes with optimal correcting properties,","cited_arxiv_id":null,"evidence_quote":"The classical Blokh-Zyablov result on concatenated codes attaining the GV bound, which the paper generalizes to EACQCs."},{"cited_title":"Bounds on the minimum distance of linear codes and quantum codes,","cited_arxiv_id":null,"evidence_quote":"Online bounds tables for best-known minimum distances of linear and quantum codes, used as the baselines for the claimed parameter improvements."}],"review_version":1}