{"id":"e2d9afa1-c56c-48ea-b230-412cd58b0dda","arxiv_id":"2607.16069","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A computer census classifies complex generalised weighing matrices with primitive monomial symmetry of rank at most 5 and degree up to 80, producing new matrices, Hamming-scheme families, and quantum stabiliser codes.","lead":"This paper classifies highly symmetric 'weighing matrices'—square grids of zeros and complex roots of unity with mutually orthogonal rows—by searching the centraliser algebras of monomial group actions up to degree 80. It finds new matrices, including previously open cases, and turns them into quantum error-correcting codes with exactly known minimum distances.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exhaustiveness of the census rests on unverified Schur-cover and primitive-group database calls; the admitted n=64 failure shows the toolchain is not reliable, and no code/census is deposited.","rationale":"The theoretical framework — Proposition 2.5, the use of Schur covers, the character-table computation, and the disjoint-support argument showing entry sets are roots of unity — is internally consistent. The main theorem's proof is a computation, not a derivation, so its validity is tied to the correctness of the computational pipeline. The paper's own statement that n=64 could not be handled is a red flag, since the same pipeline is used for all n≤80. Without a public census and code, or an independent recomputation, the exhaustiveness claim cannot be verified from the text. The reader's CONDITIONAL verdict is appropriate; I do not see a mathematical flaw in the proof of Proposition 2.5 or in the coefficient enumeration that would change that verdict. If the independent recomputation matches, the concern is resolved and the classification can be trusted. If it does not, the theorem's completeness claim fails.","tokens_in":17750,"tokens_out":28312,"duration_ms":231766,"concrete_test":"Recompute the census independently with a different software stack: in GAP, enumerate all primitive groups of degree ≤80 via PrimitiveGroup, compute Schur covers using EpimorphismSchurCover or the GAP Character Table Library, and run Algorithm 2.4 (all linear characters of the point-stabilizer preimage, all α∈{0}∪⟨ζ_k⟩^r). Compare the extended-equivalence classes with Tables 2–4. Any missing or extra class would invalidate the completeness statement of Theorem 3.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 3.1) is that the census is complete for indecomposable CGW matrices with primitive strong automorphism projection of rank r≤5 and degree n≤80, n≠64, and orbital coefficients in {0}∪⟨ζ_k⟩, k≤6. Completeness rests entirely on Algorithm 2.4, Step 1: for every primitive group G of degree ≤80, the MAGMA/SAGEMATH/GAP code must (i) enumerate G, (ii) compute a correct Schur cover bG, and (iii) form the full preimage bH of a point stabilizer. Section 3 explicitly states that at n=64 'standard routines in SAGE and MAGMA did not return Schur covers' and the search is 'therefore incomplete' — an admitted failure of exactly this step. The paper gives no evidence that the routines succeeded for every n≠64: no verification logs, no deposited code or census ('available from the authors on request'), and the two independent implementations share ATLASREP for almost-simple covers, so a database bug could affect both. If any primitive group of degree ≤80 is absent, or any computed 'Schur cover' is not a stem cover (or the point-stabilizer preimage is wrong), Proposition 2.5's sounding theory does not repair the gap: the exhaustive search would omit classes, and Theorem 3.1's 'then W is monomially equivalent to a matrix in our census' would be false. The abstract further overstates the result by omitting the n≠64 caveat and the indecomposable condition (the latter is implied by primitivity, but the former is not).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a centraliser-algebra framework for complex generalised weighing matrices (CGW matrices). For primitive permutation groups of degree n ≤ 80 (excluding n = 64) and rank 2–5, and for coefficient orders k ≤ 6, it runs a computer search over linear characters of Schur covers and point-stabiliser preimages, enumerating orbital coefficient vectors and recording matrices whose eigenvalue vectors have constant modulus. The main claim (Theorem 3.1) is that every indecomposable CGW matrix whose unimodular strong automorphism group projects onto such a primitive group and whose orbital coefficients can be rescaled into {0} ∪ ⟨ζ_k⟩ is monomially equivalent to a matrix in the authors' census. The paper also gives a Lam–Leung non-existence criterion, classifies flag-transitive and two-summand weighing matrices in Hamming schemes (Theorems 3.5 and 3.6), reports new isolated matrices including CGW(15,7;3), CGW(10,7;6), a symmetric W(21,9), and a skew W(100,36), and constructs quantum stabiliser codes with claimed exact minimum distances.","tokens_in":18117,"tokens_out":11526,"duration_ms":118402,"significance":"If the census is correct, Theorem 3.1 is a substantial computational classification covering a natural class of CGW matrices, and the paper's theoretical contributions — especially the Schur-cover reduction (Proposition 2.5), the Lam–Leung obstruction (Proposition 2.7), and the Hamming-scheme classifications (Theorems 3.5 and 3.6) — are interesting and appear sound. The exact minimum-distance computations in Section 4 are a strength, as is the explicit algebraic construction of a [[25,17,3]]_2 code matching best-known parameters. However, the central completeness claim is not independently checkable: no code, census, verification logs, or software-version information is deposited, and the abstract states a broader theorem than Theorem 3.1 actually proves.","major_comments":[{"comment":"The completeness claim in Theorem 3.1 is load-bearing and currently rests on unverifiable computation. Algorithm 2.4 Step 1 requires enumerating every primitive group of degree ≤80 (except 64) and computing a correct Schur cover and point-stabiliser preimage; Section 3 admits the toolchain fails at n=64. No census, code, verification logs, or version information is deposited (the Data Availability section says 'available from the authors on request'), and the two independent implementations share ATLASREP for almost-simple covers, so a database bug could affect both. If any primitive group or cover is missing, Proposition 2.5's theory does not repair the search. Please deposit the census and scripts, with exact software versions and a per-group log of successful Schur-cover and preimage computations, or weaken Theorem 3.1 to a conditional computational result.","section":"Algorithm 2.4 / Theorem 3.1"},{"comment":"The abstract claims a classification for degree at most 80 and coefficient orders k≤6 without the n≠64 exception, the indecomposability hypothesis, or the coefficient-condition restriction in Theorem 3.1. Section 3 states that the degree-64 search is incomplete, so this is not a harmless abbreviation: the abstract states a stronger result than is proved. Moreover, Theorem 3.1 condition 2 restricts the orbital coefficients, not merely the entry phases; the introduction and Section 3 should state this limitation explicitly. Please revise the abstract and Section 1 to match the precise scope of Theorem 3.1.","section":"Abstract / Theorem 3.1"},{"comment":"The exactness of every reported quantum code minimum distance is a central advertised feature, but the verification methods for q ∈ {5,9} rely on orbit computations under monomial automorphism groups and on the MacWilliams transform; no code or logs are provided. In particular, the claim that the orbit-reduced enumeration is exhaustive is not checkable. This does not appear to be a mathematical error, but it is part of the same reproducibility gap. Once the census and code are deposited, these computations should be included or at least checksummed.","section":"Section 4.2 / Table 1"}],"minor_comments":[{"comment":"Typo: 'from the these matrices' should read 'from these matrices'.","section":"Abstract"},{"comment":"The sentence 'checking, for every proper divisor k′ of k' is ambiguous: if the label is the minimal entry phase, the divisors should be of K = lcm(k, ord(χ)), not of k. Please clarify whether 'minimal phase' refers to coefficients or to entries.","section":"Section 3, minimal-phase paragraph"},{"comment":"Section 1 reports '136 equivalence classes' while Section 5 reports '126 extended equivalence classes'. The appendix says ten pairs of classical classes merge under extended equivalence, so the numbers are consistent, but the terminology should be made explicit at first use.","section":"Section 1 vs. Section 5"},{"comment":"The tables would be much more useful if the appendix identified which rows correspond to decomposable/direct-sum matrices and which rows are new versus previously known; currently this information is only in prose in Section 5.","section":"Section 3 / Tables 2–4"}],"recommendation":"major_revision","confidential_remarks":"I would not accept the paper in its present form because its main classification theorem cannot be independently verified and the abstract overstates its scope. The mathematical core appears sound, and the Hamming-scheme theorems and exact-distance computations are valuable. The paper is likely acceptable after the authors deposit the census and code, provide verification logs, and correct the abstract/introduction to state the n≠64 and coefficient-condition caveats."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper is a genuine advance in the study of complex generalised weighing matrices. The key theoretical result, Proposition 2.5, is correct: any indecomposable CGW whose unimodular strong automorphism group projects onto a transitive group G actually sits, up to monomial equivalence, in the centraliser algebra of a linear character of a Schur cover of G. This is a real theorem, and it does not need the perfectness assumptions in the earlier Hadamard paper. The Hamming scheme analysis (Theorems 3.5 and 3.6) is also mathematically solid, completely pinning down the flag-transitive weighing matrices on H(d,q) and the two-summand case for d=3. Those are checkable from the text.\n\nThe census itself is where I get cautious. The exhaustiveness of Theorem 3.1 rests entirely on Algorithm 2.4 running correctly on every primitive group of degree ≤80 except 64. The paper admits that at degree 64 the standard MAGMA/SAGE routines did not return Schur covers, so the search is incomplete there. For the rest, we are asked to trust that the group libraries, Schur covers, and point-stabiliser preimages are all correct—but no code, verification logs, or the census itself are deposited. 'Available from the authors on request' is not good enough for a claim of classification. This is not a flaw in the mathematics, but it is a hole in the evidence. If a group or cover is missing, Theorem 3.1 is false as stated.\n\nThere is also a small overstatement in the abstract: it says 'degree at most 80' without the 'n ≠ 64' caveat. The body is honest, but the abstract oversells.\n\nThe new matrices look plausible: CGW(15,7;3), CGW(10,7;6), the symmetric W(21,9), the skew W(100,36) from the Hall–Janko graph, and the quantum code table. Recoveries of known families are properly credited, and the Hamming-scheme family is a nice infinite family.\n\nBottom line: this deserves a serious referee. The theory is strong and the reported results are likely correct, but the computational census must be independently checkable. I would send it to a competent algebraic-design theorist with the explicit requirement that the authors deposit the code and census before acceptance. If they do that, this is a solid paper. If not, the completeness claim should be softened to a 'computational search' rather than a 'classification'.","headline":"Solid theoretical machinery and a plausible census, but the completeness claim rests on an undocumented computation; worth refereeing with code deposit required.","tokens_in":124,"tokens_out":2619,"would_cite":true,"duration_ms":28603,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B20","20C25","94B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every high-symmetry weighing matrix up to order 80 classified","keywords":["complex generalised weighing matrix","monomial representation","centraliser algebra","Schur cover","primitive permutation group","Hamming scheme","quantum error-correcting code","Hall-Janko graph"],"falsifier":"Find an indecomposable complex generalised weighing matrix of order at most 80 (other than 64) with a primitive rank-≤5 monomial strong-automorphism group and phases of order ≤6 that is not monomially equivalent to any matrix in the census, or exhibit a flag-transitive weighing matrix in a Hamming scheme H(d,q) with q odd. Alternatively, run the described character search at degree 64 with complete Schur covers; any new matrix beyond the census would falsify the stated completeness of the computational part.","tokens_in":17652,"feed_emoji":"🔢","tokens_out":5085,"duration_ms":47931,"temperature":0.7,"pith_summary":"Complex generalised weighing matrices are square arrays with zeros and roots of unity whose rows are mutually orthogonal; this paper sets out to classify every such matrix of order at most 80 whose symmetries include a primitive permutation group of rank at most 5 acting via monomial matrices, with entries in roots of unity of order at most 6. The authors prove that the classification is exhaustive: any indecomposable matrix with these symmetries lies, up to monomial equivalence, in the centraliser algebra of a monomial representation lifted from a Schur cover, so a finite search over linear characters catches everything. The resulting census of 136 equivalence classes recovers the known finite-geometry families, adds infinite families built from Hamming schemes and conference matrices, and settles the existence of several small open cases. It also yields Hermitian self-orthogonal codes and hence quantum error-correcting codes, with every minimum distance computed exactly.","feed_headline":"Every high-symmetry weighing matrix up to order 80 classified","feed_subtitle":"Results cover primitive group ranks up to 5, resolve open cases, and build quantum codes with exact minimum distances.","key_machinery":"The central object is the centraliser algebra C(ρ_χ) of a monomial representation induced from a linear character χ of a point stabiliser in a Schur cover. Its basis matrices are indexed by orientable orbitals; entries along an orbital are forced by character values, and the eigenvalue of a linear combination is read from the character table via Proposition 2.3. The defining condition WW*=wI then becomes the finite condition that every eigenvalue has modulus √w, which is checked by exhaustive enumeration over the coefficient set {0} ∪ ⟨ζ_k⟩. Two further identities carry the main results: the Hamming-scheme eigenvalue formula of Lemma 3.4 (eigenvalues of the j-th basis matrix are elementary s","core_discovery":"At the paper's centre is a completeness statement: for n ≤ 80, n ≠ 64, r ∈ {2,3,4,5}, and k ∈ {2,3,4,5,6}, every indecomposable complex generalised weighing matrix of order n whose unimodular strong automorphism group projects onto a primitive permutation group of rank r, and whose orbital coefficients can be multiplied by a global scalar into {0} ∪ ⟨ζ_k⟩, is monomially equivalent to one of the 136 matrices in the census. The key step is Proposition 2.5: because the automorphism group acts by monomial matrices, its projective representation lifts through any Schur cover, so the search over linear characters of the point-stabiliser preimage is complete even when the group is not perfect. The","pith_inferences":["Because the obstruction at degree 64 is computational, not theoretical, completing Schur-cover enumerations there would likely extend the classification; the same method should transfer to slightly larger degrees or rank 6 with more computing power.","The vanishing-sum criterion of Proposition 2.7 gives a cheap test for ruling out rank-3 candidates: for prime k, a strongly regular graph with μ not divisible by k cannot support such matrices, so it could prune future searches over larger graphs.","The Hamming-scheme theorem suggests a template for other association schemes: ask which flag-transitive bases force Kronecker-power weighing matrices, and whether the 'only if' part extends to other P- and Q-polynomial schemes.","The skew-symmetric Hall-Janko signing may be of independent interest as an orthogonal design whose underlying graph is strongly regular; analogous signings of other primitive rank-3 graphs could be searched using the same centraliser-algebra pipeline."],"forward_implications":["The classification is exhaustive in its stated range: any matrix satisfying the symmetry and phase hypotheses is monomially equivalent to a census matrix, so the appendix tables can serve as a reference for future constructions.","The existence of CGW(15,7;3), CGW(10,7;6), symmetric W(21,9), and skew W(100,36) is settled; these were open in earlier catalogues.","Flag-transitive weighing matrices in Hamming schemes H(d,q) exist only when q is even and the base is a conference matrix, yielding CGW(q^d,(q-1)^d;k); for d=3 the only sums of two basis matrices are the I+S family and two binary-cube examples.","Applying the standard Hermitian self-orthogonal code construction yields quantum codes, including an infinite [[4^d, 4^d-2^{d+1}, 3]] family and a [[25,17,3]]_2 code matching best-known parameters, all with exact distances."],"fun_headline_variants":["Complete CGW census for degree ≤ 80, rank ≤ 5","Primitive rank ≤ 5 weighing matrices up to 80: 136 classes","Weighing matrix census yields quantum codes with exact distances","All indecomposable CGWs of degree ≤ 80 and rank ≤ 5 found"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The completeness of the classification rests on the computational group libraries: if the list of primitive permutation groups of degree ≤80 (n≠64) is incomplete, or if any Schur cover or point-stabiliser preimage is returned incorrectly, then matrices satisfying the hypotheses could be missed even though Proposition 2.5 is sound; the paper itself flags that degree 64 was not completed by the standard routines.","fun_headline_variants_meta":{"raw":{"variants":["Complete CGW census for degree ≤ 80, rank ≤ 5","Primitive rank ≤ 5 weighing matrices up to 80: 136 classes","Weighing matrix census yields quantum codes with exact distances","All indecomposable CGWs of degree ≤ 80 and rank ≤ 5 found"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002344,"raw_usage":{"total_tokens":8866,"prompt_tokens":739,"completion_tokens":8127,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":8046}},"tokens_in":483,"tokens_out":8127,"duration_ms":56990,"temperature":1.0,"reasoning_tokens":8046,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T21:26:39.373004+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find an indecomposable complex generalised weighing matrix of order at most 80 (other than 64) with a primitive rank-≤5 monomial strong-automorphism group and phases of order ≤6 that is not monomially equivalent to any matrix in the census, or exhibit a flag-transitive weighing matrix in a Hamming scheme H(d,q) with q odd. Alternatively, run the described character search at degree 64 with complete Schur covers; any new matrix beyond the census would falsify the stated completeness of the computational part.","supporting_citations":[],"review_version":1}