{"id":"f7d6c737-9554-4240-bd09-4d73494fc3d9","arxiv_id":"2608.05781","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit Hermitian self-dual MDS codes prove the existence of AME(12,5), AME(18,11), AME(18,13) and the projections AME(17,11), AME(17,13).","lead":"This paper prints explicit code matrices that prove the existence of five previously open absolutely maximally entangled quantum states. The result settles listed open cases in the standard AME table using exact, checkable arithmetic.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The five AME existence claims rest on the unprinted exhaustive square-minor check for the two length-18 codes; a single unnoticed zero minor would invalidate Theorem 2.","rationale":"I read the paper in good faith: the argument is a finite explicit certificate, and the coding-theoretic reduction to Eq. (2) is standard. I checked the stated convention for F25/A5, the conjugation rules, and a few inner products; they are consistent. The minor count sum_{s=1}^9 C(9,s)^2 = C(18,9)-1 = 48,619 is correct. The one place where the proof stops being something a reader can independently verify from the printed page is the complete square-minor enumeration for the two length-18 matrices. The reader's weakest assumption identifies exactly this, and I agree. This is not an ad hominem or a disagreement with consensus; it is a concrete, finite computational assertion with no accompanying code or certificate. A single zero minor in either length-18 matrix would undo the MDS property and, through Eq. (4), the q=11/q=13 AME claims. Because the matrices are printed and the computation is finite and deterministic, an independent enumeration or a released certificate is a decisive test. I therefore leave the reader's CONDITIONAL verdict unchanged: the result should be accepted only after the enumeration is independently confirmed or a verification artifact is provided.","tokens_in":6420,"tokens_out":14455,"duration_ms":123527,"concrete_test":"Independently reconstruct A_11 and A_13 from the kernels in Eqs. (11) and (13) using the coordinate ordering in Eq. (9) and the group-circulant rule in Eq. (5). Run an exact-arithmetic enumeration of all nonempty square minors over F_121 and F_169 with independent software (e.g., SageMath, GAP, or a short custom script) and confirm all 48,619 minors per matrix are nonzero. Alternatively, ask the authors to release a machine-checkable certificate (for example, a log of pivot choices from Gaussian elimination on each submatrix) so the claim can be verified without re-running the search. If any zero minor appears in either length-18 matrix, the corresponding code and its AME consequences fail; the q=5 claim is independent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central certificate is Criterion 1: the printed A matrices must satisfy both A A^T = -I and superregularity (all nonempty square minors nonzero). The Hermitian-product identities in Eq. (15) are explicitly checkable, and I verified the q=5 spot check is consistent, so the self-duality leg is solid. The MDS leg for q=11 and q=13, however, is only asserted: the proof of Theorem 2 says 'Gaussian elimination over the same fields shows that every nonempty square minor is nonzero,' but prints no code, no log, no certificate, and no witness for the 48,619 checks per matrix (97,238 total). These two matrices are constructed from the 3x3 kernels by Eq. (5), so the enumeration is in principle reproducible, but the paper gives no verification artifact. Because AME(18,11), AME(18,13), and their projections AME(17,11), AME(17,13) all depend on those specific superregular matrices, one arithmetic error or skipped minor would destroy the central claim. This is the load-bearing unexamined input; it is a reproducibility gap, not an observed contradiction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs explicit generator matrices for Hermitian self-dual MDS codes with parameters [12,6,7]_25, [18,9,10]_121, and [18,9,10]_169. Using the standard stabilizer-to-AME correspondence, it proves the existence of AME(12,5), AME(18,11), and AME(18,13), and via one-party projection also AME(17,11) and AME(17,13). The construction is based on group-circulant kernels with Z_3^2 symmetry, discovered from a direct search at length 12. The proof relies on two exact conditions: A A^T = -I (Hermitian self-duality) and superregularity (all square minors nonzero). The self-duality identities are stated explicitly; the superregularity is asserted from a computer enumeration without a published certificate.","tokens_in":6633,"tokens_out":6394,"duration_ms":54463,"significance":"If the square-minor enumerations are correct, the paper resolves five previously open questions in the AME existence table, including two cases at q=11 and two at q=13. The group-circulant symmetry reduction is a valuable technique for future AME constructions. The paper is careful to distinguish the search from the certificate, and the printed matrices provide a concrete starting point for independent verification. The main weakness is the lack of a reproducible verification artifact for the 97,238 square-minor checks, which is essential for confidence in the theorem.","major_comments":[{"comment":"The proof's two computational claims are not independently verifiable from the manuscript. The identities in Eq. (15) are asserted as 'exact arithmetic' without showing the products, and the superregularity check is described only by the sentence 'Gaussian elimination over the same fields shows that every nonempty square minor is nonzero.' No code, log, certificate, or checksum is provided for the 923 (A_5) and 48,619 (each A_q) minor computations, nor for the finite-field arithmetic in Eq. (15). Since Theorem 2's five AME existence statements depend on these exact matrices being Hermitian self-dual and superregular, this is a load-bearing gap. In particular, a single zero minor among the 97,238 checked would invalidate AME(18,11), AME(18,13), AME(17,11), and AME(17,13). The abstract's phrase 'certified by ... complete square-minor enumeration' overstates what the paper actually supplies. Please provide a reproducible verification artifact (e.g., a short script in Sage or Python that reconstructs the matrices from Eqs. (8), (11), (13) and verifies both conditions, or a machine-checked certificate of all minor determinants). Such an artifact would allow referees and readers to confirm the theorem rather than take the assertion on faith.","section":"Section IV, proof of Theorem 2"}],"minor_comments":[{"comment":"The displayed identity 'AA T = −I k' should be 'A A^T = -I_k' with proper superscripts, and 'detA[R,C]̸= 0' should include the missing parentheses around the submatrix.","section":"Section II, Eq. (2)"},{"comment":"The convolution sum in Eq. (6) is over the group H, but the field in which the arithmetic takes place is not explicitly stated; please specify that all operations in the sum are in F_{q^2}.","section":"Section III, Eq. (6)"},{"comment":"The sentence 'related Euclidean permutation-group analysis appears in Ref. [24]' is vague; a brief indication of the specific result or connection would help the reader.","section":"Section V"},{"comment":"Several references are arXiv preprints dated 2026 (e.g., Refs. [10], [29], [39]); please confirm their accessibility and ensure the citations point to the correct versions.","section":"References"},{"comment":"Providing the three matrices in a machine-readable supplementary file would greatly facilitate independent verification, even in addition to the requested verification script.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The central question for acceptance is whether the computational verification can be made reproducible. If the authors supply a short verification script or a machine-checkable certificate for the 97,238-minor check, the paper is likely acceptable. The AI-use disclosure is transparent, but it heightens the need for a human-readable certificate that does not depend on trust in the search or verification process."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Resolving five open AME cases with explicit Hermitian self-dual MDS codes is a real result, and the paper's logic is honest: the certificate is the printed matrices plus two checks, and the search is explicitly non-exhaustive. The q=5 code is also shown non-equivalent to GRS, which is a nice bonus. I think the central claims are likely true, but the paper currently asks the reader to take a large computational assertion on faith.\n\nThe self-duality leg is fine: the matrices are printed, field conventions are stated, and I spot-checked Eq. (8) against the stated conjugation; the identity AA^T = -I can be verified by hand for the 6x6 and by nine kernel equations for the 18x18 cases. The MDS leg is where it gets soft. For A_11 and A_13 the proof says 'Gaussian elimination over the same fields shows that every nonempty square minor is nonzero,' then counts 48,619 minors per matrix, but prints no code, no log, and no certificate. A single skipped or miscalculated zero minor would kill Theorem 2. This is a reproducibility gap, not an observed contradiction, and I don't see any reason to suspect the claim is false. But in this era, a computational certificate of this size should ship the verifier, especially when the whole theorem rests on it.\n\nThe citation and context work looks careful: the authors name the corrected duadic theorem, the distance-nine codes at the same parameters, and the GRS bound, and they don't oversell classification. The explicit statement that the searches were not exhaustive is good practice. The AI-use disclosure is unusually thorough.\n\nMy bottom line: worth a serious referee. I'd send it to review but ask for the minor-enumeration code or an independently generated log before acceptance. For a reader, treat the result as conditional until that artifact appears. The proof structure is sound enough that I expect it to survive.","headline":"Resolving five open AME cases with explicit codes; credible result with one reproducibility gap in the unprinted minor enumeration.","tokens_in":7180,"tokens_out":2013,"would_cite":false,"duration_ms":17544,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Explicit Hermitian self-dual MDS codes prove five previously open AME states exist.","keywords":["absolutely maximally entangled states","Hermitian self-dual MDS codes","quantum MDS codes","stabilizer construction","group-circulant matrices","finite-Fourier transform","square-minor enumeration","Z_3^2 symmetry"],"falsifier":"Recompute, in exact arithmetic, the Hermitian products $A_5 A_5^{\\top}$, $A_{11} A_{11}^{\\top}$, and $A_{13} A_{13}^{\\top}$ from the printed matrices, reconstructing $A_{11}$ and $A_{13}$ from the kernels in Eqs. (11) and (13) via Eq. (5), and enumerate every nonempty square minor of each $A$ block. A single zero square minor in any of the three matrices would invalidate the MDS claim and hence the corresponding AME existence statement; a single failed Hermitian product would invalidate the self-duality claim.","tokens_in":6216,"feed_emoji":"⚛️","tokens_out":15143,"duration_ms":128955,"temperature":0.7,"pith_summary":"The paper proves that five absolutely maximally entangled (AME) states exist, for party numbers and local dimensions (12,5), (17,11), (18,11), (17,13), and (18,13); two of these parameters were previously listed as open. The strategy is constructive and exact. The authors exhibit three generator matrices and verify that each row space is a Hermitian self-dual maximum distance separable (MDS) code, meaning the code is its own Hermitian dual and attains the Singleton distance bound. The standard stabilizer construction converts each such code into a pure quantum MDS code, which is exactly an AME state, and projecting one party out of the even-party cases produces the two odd-party cases. Because AME states act as quantum error-correcting codes and as ingredients for quantum secret sharing and multiunitary matrices, the settled parameters become available resources.","feed_headline":"Explicit code matrices prove five AME states exist","feed_subtitle":"The construction closes open AME cases at local dimensions 5, 11, and 13.","key_machinery":"The load-bearing mechanism is the systematic generator matrix $G=[I_k|A]$ over $\\mathbb{F}_{q^2}$, together with Criterion 1: the row space of $G$ is a Hermitian self-dual $[2k,k,k+1]_{q^2}$ MDS code exactly when $A A^{\\top} = -I_k$ and every nonempty square minor of $A$ is nonzero. The first equation is the self-duality check, and the square-minor condition is systematic superregularity, which forces the MDS distance. For the length-eighteen cases, a regular $\\mathbb{Z}_3^2$ coordinate orbit forces $A_{x,y}=a(y-x)$ for $x,y\\in \\mathbb{Z}_3\\times\\mathbb{Z}_3$, so the $9\\times9$ block is determined by the nine-entry kernel $a$; self-duality becomes the convolution equation $\\sum_{h} a(h+g)\\overline{a(h)}=-\\delta_{g,0}$, which the finite-Fourier transform separates into independent character blocks. The final certificate is purely arithmetic: exact products and complete enumeration of square minors, 923 for the $6\\times6$ matrix and 48,619 for each $9\\times9$ matrix.","core_discovery":"The paper's central claim is Theorem 2: the matrices in Eqs. (8), (11), and (13) define Hermitian self-dual MDS codes with parameters $[12,6,7]_{25}$, $[18,9,10]_{121}$, and $[18,9,10]_{169}$, and consequently the AME states in Eq. (1) exist. The bridge is exact: an $\\mathbb{F}_{q^2}$-linear Hermitian self-dual MDS code of length $2k$ and dimension $k$ maps through the stabilizer construction to a pure quantum MDS code $[[2k,0,k+1]]_q$, which is an $\\mathrm{AME}(2k,q)$ stabilizer state; the reduction $\\mathrm{AME}(2k,q) \\Rightarrow \\mathrm{AME}(2k-1,q)$ by projecting any one party yields the two odd-party states. The length-eighteen matrices were not found by brute force over all $9\\times 9$ blocks: imposing a regular $\\mathbb{Z}_3^2$ coordinate orbit forces a group-circulant form determined by a nine-entry kernel, and a finite-Fourier transform separates the self-duality equations before the MDS minor checks are run.","pith_inferences":["If the length-eighteen certificates hold, the same $\\mathbb{Z}_3^2$ group-circulant reduction could be tried for other groups and finite fields, making Hermitian self-dual MDS searches feasible at parameters where full $9\\times9$ brute force is not.","The paper separates the search from the certificate, so a skeptical reader can verify the theorem directly from the printed matrices without reproducing the automorphism computation.","A natural follow-up is to test the two length-eighteen codes for monomial equivalence against each other and against other distance-ten constructions, parallel to the paper's non-GRS check for the $q=5$ code."],"forward_implications":["The five AME parameters in Eq. (1) are removed from the open cases, so protocols needing an AME(12,5), AME(17,11), AME(18,11), AME(17,13), or AME(18,13) state have an explicit stabilizer construction to use.","For $q=11$ and $q=13$ there now exist quantum MDS codes $[[18,0,10]]_q$ and, by projection, $[[17,0,9]]_q$, reaching the maximum distance allowed by the quantum Singleton bound.","The $q=5$ length-twelve code is not monomially equivalent to a generalized Reed–Solomon code, so the existence proof genuinely goes outside that classical family.","The certificate does not classify the states, so local-unitary and stochastic local operations with classical communication (SLOCC) equivalence classes among these constructions remain open questions."],"supporting_citations":[{"why":"supply the systematic generator-matrix criterion used in Criterion 1, connecting Eq. (2) to Hermitian self-duality and MDS distance.","marker":"[13, 14]"},{"why":"establish the nonbinary stabilizer construction that converts a Hermitian self-dual MDS code into a pure quantum MDS code and hence an AME state.","marker":"[15, 16]"},{"why":"gives the theorem AME(2k,q) implies AME(2k-1,q) by one-party projection, used for the odd-party states.","marker":"[2]"},{"why":"provides the online AME existence table that lists two of the settled parameters as open, defining the paper's target.","marker":"[11]"},{"why":"bounds the Schur-square dimension of generalized Reed–Solomon codes, used to show the q=5 code is not monomially equivalent to that family.","marker":"[26]"},{"why":"reports earlier Hermitian self-dual codes at the length-eighteen parameters with distance nine, the baseline the new distance-ten constructions improve.","marker":"[30]"},{"why":"corrects the earlier duadic self-duality claim and shows the required duadic splitting fails, explaining why the q=13 length-eighteen case was unresolved.","marker":"[28]"}],"fun_headline_variants":["New codes solve five AME existence problems","Explicit constructions confirm five AME states","Symmetry-guided codes prove five AME states exist","Group-circulant matrices settle five AME cases","Hermitian self-dual codes yield five new AME states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the complete enumeration of all 48,619 square minors of each length-eighteen matrix was performed correctly; the paper asserts the computation but provides no code, log, or independent certificate.","fun_headline_variants_meta":{"raw":{"variants":["New codes solve five AME existence problems","Explicit constructions confirm five AME states","Symmetry-guided codes prove five AME states exist","Group-circulant matrices settle five AME cases","Hermitian self-dual codes yield five new AME states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000302,"raw_usage":{"total_tokens":1744,"prompt_tokens":956,"completion_tokens":788,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":713}},"tokens_in":572,"tokens_out":788,"duration_ms":6425,"temperature":1.0,"reasoning_tokens":713,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:13:27.284388+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute, in exact arithmetic, the Hermitian products $A_5 A_5^{\\top}$, $A_{11} A_{11}^{\\top}$, and $A_{13} A_{13}^{\\top}$ from the printed matrices, reconstructing $A_{11}$ and $A_{13}$ from the kernels in Eqs. (11) and (13) via Eq. (5), and enumerate every nonempty square minor of each $A$ block. A single zero square minor in any of the three matrices would invalidate the MDS claim and hence the corresponding AME existence statement; a single failed Hermitian product would invalidate the self-duality claim.","supporting_citations":[{"cited_title":"Huber and N","cited_arxiv_id":null,"evidence_quote":"provides the online AME existence table that lists two of the settled parameters as open, defining the paper's target."},{"cited_title":"Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem","cited_arxiv_id":"2104.05122","evidence_quote":"reports earlier Hermitian self-dual codes at the length-eighteen parameters with distance nine, the baseline the new distance-ten constructions improve."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"corrects the earlier duadic self-duality claim and shows the required duadic splitting fails, explaining why the q=13 length-eighteen case was unresolved."}],"review_version":2}