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Complex generalised weighing matrices in centraliser algebras of monomial representations

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Every high-symmetry weighing matrix up to order 80 classified

desk verdict Solid theoretical machinery and a plausible census, but the completeness claim rests on an undocumented computation; worth refereeing with code deposit required. read the letter →

arxiv 2607.16069 v1 pith:LHELQGD5 submitted 2026-07-17 math.CO

Ronan Egan , Padraig Ó Catháin , Andrea Švob This is my paper · ORCID
classification math.CO MSC 05B2020C2594B05
keywords complexgeneralisedweighingmatrixmonomialrepresentationcentraliseralgebraSchurcoverprimitivepermutationgroupHammingschemequantumerror-correctingcodeHall-Jankograph
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Algorithm 2.4 / Theorem 3.1] 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.
  2. [Abstract / Theorem 3.1] 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.
  3. [Section 4.2 / Table 1] 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.
minor comments (4)
  1. [Abstract] Typo: 'from the these matrices' should read 'from these matrices'.
  2. [Section 3, minimal-phase paragraph] 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.
  3. [Section 1 vs. Section 5] 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.
  4. [Section 3 / Tables 2–4] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the classification is an independent exhaustive search with explicit completeness caveats, not a derivation from its own inputs.

full rationale

The claimed derivation chain is not circular. Proposition 2.5, the key exhaustiveness step, is proved from Schur's lifting theorem: an indecomposable CGW matrix with unimodular strong automorphism group projecting to G yields a projective representation, which lifts to a linear representation of a Schur cover, and hence to a monomial representation induced from a linear character of the point-stabilizer preimage. This is an external theorem, not an assumption of the target classification. Algorithm 2.4 then enumerates a finite coefficient set and verifies the weighing-matrix condition WW*=wI by checking that the character-table eigenvalues have constant modulus; no coefficient is fitted to data and then renamed a prediction. The classification is anchored to independent external results by recovering the Seberry-Whiteman, Berman, Goldberger, and Moorhouse families, and the minimum distances of the quantum codes are computed by separate enumeration or orbit-reduced search, not by the construction itself. The admitted failure at degree 64, where standard MAGMA/SAGE routines did not return Schur covers, is explicitly flagged as an incompleteness caveat; it weakens the scope of Theorem 3.1 but does not make the theorem follow from its inputs. The self-citations, chiefly to [1] for the centraliser-algebra framework and to [8] for the standard stabiliser-code construction, are parameter-free published results with stated assumptions that do not include the present classification; they are load-bearing but not circular in the sense of being unverified assumptions equivalent to the claimed output. No circular step of any of the enumerated kinds is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's contribution rests on standard representation-theoretic facts and on unverified-but-stated database/tool assumptions; it introduces no new physical entities or fitted parameters.

assumptions (5)
  • standard math Schur's covering theorem: every projective representation of a finite group lifts to a linear representation of a Schur cover.
    Used in Proposition 2.5 to lift the projective action of the strong automorphism group to a monomial representation of the Schur cover; cited to [12].
  • domain assumption The centraliser-basis and character-table formulas of [1] (Proposition 2.1, Theorem 2.2, Proposition 2.3) are correct.
    The paper reproduces these statements but does not reprove them; the entire search and eigenvalue criterion depend on them.
  • domain assumption MAGMA/SAGEMATH/GAP primitive-group catalogues and Schur-cover implementations are complete and correct for degree ≤80 (except 64).
    Algorithm 2.4 Step 1 iterates over these libraries; Section 3 admits failure at degree 64 for affine groups.
  • standard math Lam–Leung theorem on vanishing sums of roots of unity (Theorem 2.6).
    Used in Proposition 2.7 to derive nonexistence of rank-3 CGW matrices from the strongly regular graph parameter μ; cited to [14].
  • domain assumption nauty produces correct and complete canonical forms for directed graph isomorphism.
    Equivalence-class reduction in Section 2.2 relies on nauty canonical labels; cited to [16].

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Pith. "Pith review of Complex generalised weighing matrices in centraliser algebras of monomial representations." pith.science (2026). https://pith.science/paper/LHELQGD5

@misc{pith2026260716069,
  author       = {Pith},
  title        = {Pith review of: Complex generalised weighing matrices in centraliser algebras of monomial representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LHELQGD5}},
  note         = {Machine review of arXiv:2607.16069}
}
abstract

An $n \times n$ matrix $W$ with exactly $w$ non-zero entries taken from the set of $k^{\rm th}$ complex roots of unity in each row and column satisfying $WW^{\ast} = wI_n$ is a complex generalised weighing matrix $CGW(n,w;k)$. We study such matrices through the centraliser algebras of monomial representations of finite groups. Using an exhaustive search over the linear characters of Schur covers, we classify, up to monomial equivalence, the complex generalised weighing matrices admitting a primitive group of rank at most five and degree at most $80$ acting by strong automorphisms, for coefficient orders $k \leq 6$, with partial results for larger degrees $100$. The census recovers known infinite families related to projective and affine finite geometries, describes infinite families related to Hamming schemes and settles the existence of some small open cases enumerated in the literature. We construct quantum error-correcting codes from the these matrices and determine their minimum distances exactly in all cases.

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Reference graph

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