{"id":"bff1b3e9-b332-4376-a3f7-00bb48dd7d55","arxiv_id":"2501.15465","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"EAQECC parameters are translated into additive-code terms and several families of single-logical-qubit EAQECCs are claimed to saturate the EA-Singleton bound.","lead":"The paper recasts entanglement-assisted quantum error-correcting codes in the language of additive codes over GF(4) and uses this to construct families of optimal codes. The authors claim to settle an open optimality question for a [[5,2,d;3]] code and to disprove a conjecture about alphabet-size limits for such codes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The asserted weight enumerator for Theorem 3 item 2 is false: the codeword r1+r2+r3 has Hamming weight n, not weight 4 or n−2, so the minimum-distance proof fails as written.","rationale":"The reader's weakest assumption is exactly where the proof fails. I checked the construction: G_{3,n} has three rows, and the sum of all three rows is a nonzero codeword of weight n. The asserted enumerator contains only degrees 4 and n−2; hence it is false for every n ≥ 5 in the family. The n = 5 case is the clearest because the code is the [5, 3, 3] MDS code whose enumerator is 1 + 30z^3 + 15z^4 + 18z^5, not 1 + 15z^4 + 48z^3. This is an internal inconsistency, not a disagreement with consensus. It is load-bearing because the advertised minimum distance d_ea = n−2 is read off from the difference of the two enumerators. If the minimum distances are nevertheless correct, they require a new proof; as written, the main parameter families are unsupported. I do not rely on the apparent k/c mismatch, because that is resolved by taking the dual C⊥t as the stabilizer; the weight enumerator is the real weak point. No change to the REJECT verdict is needed.","tokens_in":8987,"tokens_out":15773,"duration_ms":141079,"concrete_test":"Independently enumerate all 4^3 = 64 codewords of G_{3,n} for n = 5 and n = 7 and compare the Hamming weight distribution with W(t) = 1 + 15z^4 + 48z^{n−2}. For n = 5 the distribution is 1 + 30z^3 + 15z^4 + 18z^5, and for n = 7 the single codeword r1 + r2 + r3 already has weight 7. If the stated W(t) is reproduced, the objection is withdrawn; otherwise Theorem 3 item 2 needs a corrected weight enumerator before its distance claim can be accepted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3, item 2 (Section 4) states that for n = 5 + 2m the generator matrix G_{3,n} produces a code Cn with weight enumerator W(t) = 1 + 15z^4 + 48z^{n−2} and radical enumerator WR(t) = 1 + 15z^4, and concludes that Cn has minimum distance n−2 outside the radical. This is contradicted by an explicit codeword. Let r1, r2, r3 be the three rows of G_{3,n}. Their sum is v = (1, 1, ϖ, 1, ϖ, 1, ..., 1), where the last 2m coordinates are 1. The Hamming weight of v is 5 + 2m = n. Because n is neither 4 nor n−2 for odd n ≥ 5, the asserted W(t) cannot be the weight enumerator of Cn. For n = 5 the actual enumerator is 1 + 30z^3 + 15z^4 + 18z^5 for the [5, 3, 3] MDS code, not the stated 1 + 15z^4 + 48z^3. Thus the proof's claimed distance d_ea = n−2 is not established by the manuscript. The central parameter families in Theorem 3 therefore rest on an unsupported weight-enumerator computation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an additive-code formulation of the entanglement-assisted stabilizer formalism via the symplectic geometry of F_2^{2n} and the trace inner product over F4, and uses it to construct four families of binary EAQECCs that are claimed to saturate the EA-Singleton bound. It also claims to settle the optimality of the [[5,2,3;3]] code by proving the nonexistence of [[5,2,4;3]], and to disprove a conjecture on the parameters of EAQMDS codes. The central constructions are proved by asserting weight enumerators of the relevant additive codes and then applying Theorem 2 to obtain the EA code parameters.","tokens_in":1645,"tokens_out":2324,"duration_ms":141438,"significance":"If correct, the additive-code reformulation would provide a convenient bridge between symplectic geometry and EAQECC parameters, and the constructed families, especially the [[n,1,n-2;n-5]] and [[n,1,n-2s;n-4s-1]] families, would be new parameter records saturating the EA-Singleton bound. The optimality proof for [[5,2,3;3]] would close an open problem listed in the literature. However, the paper's advertised disproof of the EAQMDS conjecture is not actually presented, and the main construction proofs rely on weight enumerators that are asserted without derivation, one of which is demonstrably false. As written, the central results are therefore not established.","major_comments":[{"comment":"The weight enumerator W(t)=1+15z^4+48z^{n-2} claimed for Cn is false. For the three rows r1,r2,r3 of G3,n, the F4-linear combination r1+r2+r3 equals (1,1,ϖ,1,ϖ,1,...,1) with Hamming weight 5+2m=n. For n=5 this codeword has weight 5, but the asserted enumerator has no z^5 term; the actual enumerator of the [5,3,3] code includes a z^5 term. Consequently the minimum-distance computation that yields d_ea=n-2 is unsupported, and the claimed [[n,1,n-2;n-5]] family is not proved by the manuscript's argument.","section":"Section 4, Theorem 3 item 2"},{"comment":"The weight enumerators for the codes in items 3 and 4 are not computed; they are written as WR(t)=1+a_4z^4+...+a_{8s}z^{8s} (or with 8s+4) plus a single high-weight term, with the coefficients a_i left unspecified. The sentence 'It is easy to verify' provides no derivation, and in light of the false enumerator in item 2, these asserted enumerators cannot be taken on faith. The claimed minimum distances for these families are therefore not established.","section":"Section 4, Theorem 3 items 3 and 4"},{"comment":"The paper advertises a disproof of the conjecture that EAQMDS codes satisfy a length constraint analogous to n ≤ q^2+d-2. The introduction states that the paper will show this conjecture is incorrect, and the conclusion repeats that a conjecture has been disproven, but no theorem or example in the body states the conjecture precisely or demonstrates a code that violates it. This is a promised central contribution that is missing from the manuscript.","section":"Introduction (Problem (b)) and Conclusion"},{"comment":"The nonexistence proof for [[5,2,4;3]] rests entirely on the cited classification in [27] that the only (5,2^4,4) additive MDS code has type (4,0)[t]. This type assertion is load-bearing for the optimality claim, but the paper does not reproduce the classification or provide the generator matrix of the unique code so the reader can verify the type under the paper's own definitions. As written, the proof is a black-box citation for a critical step.","section":"Example 2"},{"comment":"The proofs never explicitly identify which code is the EA-stabilizer and which is the normalizer when applying Theorem 2. For example, in item 1 the code Cn=[n,2]4 has additive type (m,c)=(4,1); if Cn were the stabilizer, Theorem 2 would give k=n+c-m=n-3 and c=1, not the claimed [[n,1,n-1;n-3]]. The claimed parameters follow when the stabilizer is Cn⊥t, with Cn serving as the normalizer, but the text says only 'Cn normalizes' and does not define this switch. This ambiguity makes the proof difficult to verify and should be corrected.","section":"Section 4, application of Theorem 2"}],"minor_comments":[{"comment":"The title reads 'optimal EAQECC codes'; this is redundant since EAQECC already contains 'codes'.","section":"Title"},{"comment":"The statement 'G2,n generates a Cn=[n,2,4] linear code' should be qualified for n≥6, since for n=4 the same construction yields a [4,2,3] code, as stated earlier in the same paragraph.","section":"Section 4, item 1"},{"comment":"The text says G4,5 generates a (5,2^4,3) code and then considers n=5+2m>5; the reader is left to infer how the (5,2^4,3) base case relates to the weight enumerator stated for the larger n, and this connection should be made explicit.","section":"Section 4, item 1 odd case"},{"comment":"There are typographical inconsistencies in notation, such as the use of both ϖ and its ASCII rendering, and the generator matrices contain apparent formatting artifacts (e.g., the '11110' row in G2,4) that should be cleaned up.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's main construction proofs are not reliable as written: one claimed weight enumerator is contradicted by an explicit codeword, and the other enumerators are simply asserted. The authors will need to recalculate or rigorously derive the weight enumerators and restructure the proofs of Theorem 3. In addition, the advertised disproof of the EAQMDS conjecture is absent, so that claim should either be added or removed from the abstract and introduction. The paper also leans on a substantial number of self-citations and an unpublished reference ([29]) for the claim of 'over 60 optimal EAQECCs', which is not verifiable from the manuscript. The proposed additive formalism itself is acknowledged to be equivalent to existing frameworks, so the contribution stands or falls on the code constructions and the optimality analysis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper's central claim—new families of optimal EAQECCs—does not survive contact with its own generator matrices. In Theorem 3, item 2, the asserted weight enumerator W(t)=1+15z^4+48z^{n-2} for the code generated by G_{3,n} is false. For n=5, the sum of the three rows is (1,1,ϖ,1,ϖ), a weight-5 codeword, and the code is the [5,3,3] MDS code with 30 weight-3 codewords. Its actual enumerator is 1+30z^3+15z^4+18z^5. So the claimed minimum distance n-2 outside the radical is not established; the actual minimum distance is 3. Items 3 and 4 follow the same block-matrix pattern and their enumerators are asserted without proof; given the concrete failure in item 2, those claims are not credible as written.\n\nWhat is genuinely of interest: the additive EA formalism in Theorem 2 is a clean restatement, though the authors acknowledge it is equivalent to earlier symplectic formulations. The non-existence argument for [[5,2,4;3]] in Example 2 is a serious, plausible use of the classification of additive MDS codes from Ball et al. That fragment may be correct and publishable on its own.\n\nThe rest of the advertised contributions are unsubstantiated. The 'over 60 optimal codes' are promised in a future paper [29], not in this manuscript. The disproof of the alphabet-size conjecture is stated about a bound that the paper never states precisely; the reader has to infer which inequality is being violated, and the construction that supposedly violates it is one of the faulty families. The paper would need major revision to make the enumerators verifiable and correct, and the families as parameterized do not exist with the claimed distances.\n\nRecommendation: desk reject in current form. The formalism restatement and the [[5,2,4;3]] non-existence example are worth a short note, but the quantum code constructions are load-bearing and wrong.","headline":"The main EAQECC families are invalid: a weight enumerator in Theorem 3 is false, leaving only the formalism restatement and a possibly correct non-existence example.","tokens_in":9829,"tokens_out":14753,"would_cite":false,"duration_ms":111511,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P70","94B05","94B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper develops a symplectic-geometric additive-code formalism for entanglement-assisted quantum codes and uses it to construct Singleton-saturating families and resolve an open optimality question.","keywords":["additive codes","quantum codes","entanglement-assisted quantum codes","symplectic geometry","optimal codes","EA-Singleton bound","quaternary additive codes","EA stabilizer codes"],"falsifier":"Compute the weight enumerator of the [5,3,3]_4 code generated by G_{3,n} with n=5. The row sum r_1+r_2+r_3=(1,1,ϖ,1,ϖ) has weight 5, whereas the asserted enumerator W(t)=1+$15z^{4}$+$48z^{3}$ contains no $z^{5}$ term; verifying this one vector would show the enumerator claim for that case fails.","tokens_in":8766,"feed_emoji":"⚛️","tokens_out":10374,"duration_ms":82790,"temperature":0.7,"pith_summary":"The paper introduces an additive-code version of the entanglement-assisted stabilizer formalism, grounded in the geometry of the symplectic group, and uses it to construct families of binary entanglement-assisted quantum error-correcting codes (EAQECCs) that saturate the EA-Singleton bound. It also settles an open optimality question by proving that no [[5,2,4;3]] code exists, so the known [[5,2,3;3]] code is optimal. The framework converts any quaternary additive code with known trace radical into an EAQECC and its dual, giving a systematic path from classical code tables to quantum codes and explaining how some EAQMDS codes can have arbitrarily large length even for binary alphabets.","feed_headline":"Symplectic geometry builds optimal entanglement-assisted quantum codes","feed_subtitle":"Additive-code formalism yields Singleton-saturating EAQECCs and proves [[5,2,3;3]] optimal.","key_machinery":"The load-bearing object is the isometry φ((a|b)) = ωa + ϖb from the 2n-dimensional binary symplectic space to F4^n. It carries a subspace of symplectic type (m, c)[s] to an (n, 2^m) additive code of type (m, c)[t], and it carries the symplectic radical V ∩ $V^{{⊥s}}$ to the trace radical R(C) = C ∩ $C^{{⊥t}}$, whose dimension is m − 2c. Theorem 2 is the engine: it says such a C EA-stabilizes [[n, n − c − l, d; c]] with d = min{wt(g) : g ∈ $C^{{⊥t}}$ \\ R(C)}, and that $C^{{⊥t}}$ EA-stabilizes the dual code. The constructions in Theorem 3 then choose explicit generator matrices whose weight enumerators are simple enough to read off d directly.","core_discovery":"The central claim is that the EA stabilizer formalism can be re-framed as a statement about additive codes over F4. Given an (n, 2^m) additive code C of type (m, c)[t] with trace radical R(C) = C ∩ $C^{{⊥t}}$ of dimension l, C EA-stabilizes an EAQECC with parameters [[n, n − c − l, d_ea; c]], where d_ea is the smallest symplectic weight in $C^{{⊥t}}$ \\ R(C), and the trace dual $C^{{⊥t}}$ stabilizes the dual EAQECC with the roles of C and $C^{{⊥t}}$ exchanged. This reduces code construction to choosing additive codes with controlled trace radicals, and the paper uses it to produce the four families of Theorem 3 and to prove the non-existence of a [[5,2,4;3]] code, thereby showing the known [[5,2,3;3]] code is optimal.","pith_inferences":["Beyond the paper, the weight-enumerator route suggests that any additive code whose weight distribution is concentrated on two values will automatically normalize a high-distance EAQECC, so the search for new families can focus on two-weight additive codes.","The non-existence argument for [[5,2,4;3]] uses the classification of MDS additive codes; the same strategy could decide optimality for other small EAQECC parameters once analogous classifications exist.","The paper shows n can be arbitrarily large for binary EAQMDS codes of distance growing with n; whether this extends to larger alphabets or to EAQMDS codes with more logical qubits is left open.","The formalism is stated for binary EAQECCs, but the same additive-code translation should transfer to non-binary alphabets by replacing F4 with other finite fields and the trace inner product with the appropriate trace form."],"forward_implications":["Every quaternary additive code C with known trace radical yields an EAQECC and its dual with the parameters in Theorem 2, so existing tables of additive codes become construction engines for EAQECCs.","The families in Theorem 3 give binary EAQECCs with [[n,1,n−1;n−3]] and [[n,1,n−2;n−5]] parameters that saturate the EA-Singleton bound; the first family is EAQMDS.","The proof that no [[5,2,4;3]] code exists fixes the minimum distance of the n=5, k=2, c=3 case at d=3, resolving the open question raised in [20].","The claim that binary EAQMDS codes can have arbitrarily large n contradicts the previously expected n ≤ q^2 + d − 2 type constraint, so the examples bear on the general existence question for EAQMDS codes.","Because the formalism gives the EA-normalizer explicitly as the trace dual, it also supplies the data needed to design encoding and decoding circuits for the constructed codes."],"supporting_citations":[{"why":"Defines the entanglement-assisted stabilizer formalism and the EA-Singleton bound that all constructed codes are measured against.","marker":"[9]"},{"why":"Supplies the symplectic dualities and identities for EAQECCs that Theorem 2 restates in additive-code language.","marker":"[15]"},{"why":"Provides the quaternary additive-code framework, trace inner product, and type classification used throughout.","marker":"[4]"},{"why":"Gives the symplectic-space geometry and classification of subspaces by type (m,c)[s] that underlies the formalism.","marker":"[24]"},{"why":"Yields the sharpened Singleton-type bounds that define EAQMDS codes and saturation.","marker":"[19]"},{"why":"Is the source of the open [[5,2,d;3]] optimality problem and the known [[5,2,3;3]] code.","marker":"[20]"},{"why":"Classifies the short MDS additive codes used in the non-existence proof of [[5,2,4;3]].","marker":"[27]"},{"why":"Supplies small additive quaternary codes used as building blocks for example EAQMDS codes.","marker":"[25]"}],"fun_headline_variants":["Symplectic geometry yields optimal EAQECCs","Additive-code formalism nails optimal EA quantum codes","Geometry demystifies entanglement-assisted code optimality","Symplectic view proves best EAQECC parameters"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The distance claims for the new families rest on the asserted weight enumerators of the constructed additive codes; if one of those enumerators is incorrect for some n, the corresponding minimum distance is not established by this argument.","fun_headline_variants_meta":{"raw":{"variants":["Symplectic geometry yields optimal EAQECCs","Additive-code formalism nails optimal EA quantum codes","Geometry demystifies entanglement-assisted code optimality","Symplectic view proves best EAQECC parameters"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000239,"raw_usage":{"total_tokens":1459,"prompt_tokens":836,"completion_tokens":623,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":452,"completion_tokens_details":{"reasoning_tokens":562}},"tokens_in":452,"tokens_out":623,"duration_ms":6005,"temperature":1.0,"reasoning_tokens":562,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:18:12.628390+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the weight enumerator of the [5,3,3]_4 code generated by G_{3,n} with n=5. The row sum r_1+r_2+r_3=(1,1,ϖ,1,ϖ) has weight 5, whereas the asserted enumerator W(t)=1+$15z^{4}$+$48z^{3}$ contains no $z^{5}$ term; verifying this one vector would show the enumerator claim for that case fails.","supporting_citations":[{"cited_title":"Correcting quantum errors with entangle- ment,","cited_arxiv_id":null,"evidence_quote":"Defines the entanglement-assisted stabilizer formalism and the EA-Singleton bound that all constructed codes are measured against."},{"cited_title":"Dualities and identities for entanglement- assisted quantum codes,","cited_arxiv_id":null,"evidence_quote":"Supplies the symplectic dualities and identities for EAQECCs that Theorem 2 restates in additive-code language."},{"cited_title":"Quantum error correction via codes over GF(4),","cited_arxiv_id":null,"evidence_quote":"Provides the quaternary additive-code framework, trace inner product, and type classification used throughout."},{"cited_title":"Wan, Geometry of classical groups over finite fields and its applications","cited_arxiv_id":null,"evidence_quote":"Gives the symplectic-space geometry and classification of subspaces by type (m,c)[s] that underlies the formalism."},{"cited_title":"Entropic proofs of Singleton bounds for quan- tum error-correcting codes,","cited_arxiv_id":null,"evidence_quote":"Yields the sharpened Singleton-type bounds that define EAQMDS codes and saturation."},{"cited_title":"Bounds on the minimum distance of entanglement-assisted quantum codes,","cited_arxiv_id":null,"evidence_quote":"Is the source of the open [[5,2,d;3]] optimality problem and the known [[5,2,3;3]] code."},{"cited_title":"Griesmer type bounds for additive codes over finite fields, integral and fractional MDS codes,","cited_arxiv_id":null,"evidence_quote":"Classifies the short MDS additive codes used in the non-existence proof of [[5,2,4;3]]."},{"cited_title":"Small additive quaternary codes,","cited_arxiv_id":null,"evidence_quote":"Supplies small additive quaternary codes used as building blocks for example EAQMDS codes."}],"review_version":1}