Non-affine Families of 8 x 8 Complex Hadamard Matrices

Tuomo Valtonen

REVIEW 4 major objections 4 minor 25 references

Non-affine Families of 8 x 8 Complex Hadamard Matrices

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper constructs six new three-parameter non-affine families of 8x8 complex Hadamard matrices and shows, by exhaustive computer search, that representatives from each family are inequivalent to all previously known matrices.

desk verdict Real constructions, unproven newness: Theorem 5.2 misses the known T8^(1), and Prop. 4.5's identity has a variable-count error. read the letter →

arxiv 2505.22947 v2 pith:LZFGOBJZ submitted 2025-05-29 math.CO

Tuomo Valtonen This is my paper

classification math.CO MSC 05B20

keywords complexHadamardmatricesnon-affinefamiliespalindromicpolynomialsequivalenceButsonorder8unimodularparametersiterativeinversionalgorithm

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

This paper constructs six new three-parameter non-affine families of complex Hadamard matrices of order 8: $T_{8C}^{(3)}$, $T_{8D}^{(3)}$ and its transpose, $T_{8E}^{(3)}$ and its transpose, and $T_{8F}^{(3)}$. A complex Hadamard matrix here is an $8\times 8$ matrix whose entries have modulus 1 and whose rows are mutually orthogonal. Each family is given explicitly as the image of a function of three unimodular parameters, with the parameter domain chosen so that the matrix entries stay unimodular. The paper's central claim is that a representative matrix from each family is inequivalent to every matrix in the previously known affine and non-affine families, so the six families are pairwise inequivalent and contain genuinely new Hadamard matrices. The construction method, which solves the orthogonality equations through palindromic polynomials, also produces Butson matrices of order 8 that lie outside all previously listed families.

What carries the argument

The central objects are multivariate conjugate-reciprocal, or palindromic, polynomials. An orthogonality condition between two rows of a candidate matrix is a polynomial in the unimodular entries; the paper rewrites each such system so that every constraint is palindromic. Lemma 2.4 then turns a palindromic equation into sums of the form $y_j + y_j^{-1}$, which become trigonometric equations after the substitution $x_j = e^{i\xi_j}$, while Lemmas 2.5 and 2.6 give explicit conditions, for instance $|p_1|/|p_2|\le 2$, under which the solved variables are automatically unimodular. This machinery turns the six-variable orthogonality systems behind $T_{8C}, \ldots, T_{8F}$ into three-parameter families with explicit domains. On the equivalence side, the paper introduces the notion of an iteratively invertible family function: entries $\phi_1, \ldots, \phi_k$ that can be inverted one after another to recover the parameters from the matrix, enabling a backtracking search for family membership.

What would settle it

Take the representative matrices $H' = f(e^{2i},\ldots,e^{(k+1)i})$ for the six new families, together with $B_1$ and $B_2$, and rerun the membership test with an independent exact-arithmetic or high-precision equivalence check; if any of these matrices is found equivalent to a matrix from a different family in list (9), Theorem 5.2 and Proposition 5.3 would be false, and an observed misclassification at the tolerance parameter $\varepsilon$ would explain the error.

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Extended reading notes

Core claim

The paper claims that the known catalogue of order-8 complex Hadamard matrices is incomplete, and that the missing pieces are six 3-parameter non-affine families generated by the functions $S_C$, $S_D$, $S_E$, and $S_F$, with transposes counted as separate families. For each of these functions the paper gives an explicit formula for three of the six matrix-entry variables in terms of the three free parameters, and it specifies the domain on which the formulas stay unimodular. Theorem 5.2 is the load-bearing claim: for the twelve families listed in (9), the matrix $H' = f(e^{2i}, e^{3i}, \ldots, e^{(k+1)i})$ belongs to the family that defines it and to no other family in the list. The proof is an exhaustive computer search using an iterative-inversion algorithm: choose certain matrix entries that determine the parameters, invert them, and then check permutation equivalence. Because the representative $H'$ is not a boundary point, the result is not an isolated coincidence: a whole neighbourhood of the family consists of matrices new to the catalogue.

Load-bearing premise

The load-bearing premise is that the exhaustive computer search of Section 5.1, whose floating-point arithmetic, equivalence algorithm, code, data, and tolerance parameter are not provided, correctly decides all family memberships and equivalences; if the search misses one equivalence, the claimed newness and pairwise inequivalence of the families collapse.

Editorial extensions

If this is right

The six new families are pairwise inequivalent and none is contained in another, so the known list of inequivalent order-8 complex Hadamard families strictly expands.
Each representative $H'=f(e^{2i},\ldots,e^{(k+1)i})$ lies in the interior of its family's domain, so a whole neighbourhood of genuinely new matrices exists around it, not just a single matrix.
The Butson matrices $B_1, B_2 \in BH(8,6)$ and certain $BH(8,q)$ matrices with $q>6$ are not equivalent to any matrix in the twelve-family list (9).
The 1-parameter family $T_8^{(1)}$ embeds, up to transpose and equivalence, into the six-variable precursor $T_{8D}(T^6)$, consistent with the paper's suggestion that it is a subfamily of the new family $(T_{8D}^{(3)})^T$.
Every new family is given by explicit formulas for three dependent parameters, so matrices in the family can be generated directly from three unimodular parameters without solving the full orthogonality system.

Reading between the lines

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

The paper leaves open whether $B_1$ and $B_2$ are isolated; tracing their neighbourhoods could reveal still more order-8 families, since any smooth family containing them would have to contain a neighbourhood of nearby Hadamard matrices.
The same palindromic-polynomial reduction should apply to other orders and other Butson-type classes: whenever orthogonality constraints factor into palindromic polynomials, Lemmas 2.4-2.6 reduce parametrization to trigonometric inequalities.
The unresolved conjecture that $T_{8C}^{(3)}$ and $T_{8F}^{(3)}$ are symmetric suggests the true count of inequivalent new families may be smaller than six; proving or disproving symmetry would settle the final list.
An exact-arithmetic certification of the floating-point search would turn the computational inequivalence into a theorem verifiable by formal proof and would remove the dependence on the unpublished equivalence algorithm.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

My take: this is a paper with real algebraic content and a central claim that is not yet proven. The constructions—explicit closed-form parametrizations of four non-affine order-8 families via palindromic polynomial systems—are genuine work and likely useful to anyone doing classification. T8E's rational family is especially clean, and the palindromic reduction lemma is a nice tool. But the headline that these families contain matrices inequivalent to anything previously known is unsupported at the moment.

The biggest problem is that Theorem 5.2 only checks membership against the list (9), which omits the known 1-parameter family T8^(1). The one place where the paper tries to bridge that gap, Proposition 4.5, contains an identity that calls the six-variable function T8D with seven arguments. As written, the proof is uncheckable. The old T8C^(3) from [4] is likewise dispensed with by saying the new T8C "seems to contain" it. So even a perfect search over list (9) would not rule out equivalence to those two known families, and the Butson examples in Section 5.3 have the same blind spot.

A second, independent gap is the nature of the exhaustive search. The algorithm in Section 5.1 relies on an unpublished companion paper [16] for permutation equivalence, and it uses floating-point arithmetic without specifying the tolerance or how the enumeration of dephased equivalent matrices is made complete. Numerical search can be useful evidence, but it is not a proof of non-membership unless the search is certified.

There are also smaller issues: domain definitions with inconsistent variables (Theorem 4.4 writes D as a subset of T^3 with variables (d,e,f) but then uses a,b,c), and Theorem 4.10's statement says T8C where it should say T8F. These are typos, but they add friction to an already hard-to-check paper.

Who should read this? People working on order-8 complex Hadamard matrices, MUBs, or related quantum information questions. They will want the explicit families and the method. But they should not cite the newness claim yet.

My recommendation: send it to peer review, but tell the authors the acceptance path requires (1) releasing code/data for the search and the equivalence algorithm, (2) fixing Proposition 4.5 and proving or retracting the T8^(1) containment, (3) clarifying how the old T8C is handled, and (4) cleaning up the typos. The algebraic core justifies referee time; the verification gap is real but addressable.

Referee Report

4 major / 4 minor

Summary. The paper constructs six explicit three-parameter families of complex Hadamard matrices of order 8, denoted T8C, T8D, T8E, T8F and their transposes, by solving polynomial orthogonality systems whose variables are forced to be unimodular by domain conditions derived from palindromic polynomial lemmas. The author claims that these families contain matrices not equivalent to any previously known complex Hadamard matrices, and supports this with Theorem 5.2, an exhaustive computer search over a list of known affine and non-affine families, plus a separate search showing certain Butson matrices fall outside the same list. The core algebraic lemmas in Section 4 are explicit and checkable, but the newness claim depends on unreleased computational infrastructure and on containment statements that are either unproved or misstated.

Significance. If the newness claim is established, this is a meaningful contribution to the classification of complex Hadamard matrices of order 8: explicit non-affine families with the correct defect dimension are rare, and the palindromic-polynomial method gives a systematic way to produce unimodular solutions. The paper deserves credit for writing the algebraic conditions in full, for not fitting parameters to the final conclusion, and for being explicit that the proof of inequivalence is computational. However, the central claim is currently not independently verifiable from the manuscript: the exhaustive search is not reproducible, its equivalence oracle is an unpublished reference, and the comparison list omits known non-affine families that the paper itself discusses.

major comments (4)

[§5.2, Theorem 5.2] The central newness claim rests entirely on an exhaustive computer search whose inputs and correctness are not documented. The proof says only that an exhaustive search with the algorithm of Section 5.1 yields the result, but Section 5.1 uses floating-point arithmetic with an unnamed tolerance ε and calls an unpublished equivalence algorithm [16] for both Step 1 and Step 3. No code, search data, tolerance value, or completeness proof for the equivalence test is provided, so the assertion that the only family containing H' is its own family cannot be independently checked. This is load-bearing for the abstract's claim that the families contain matrices not equivalent to any previously known matrices.
[§3.3 and §4.2, Proposition 4.5] The comparison list (9) in Theorem 5.2 omits the previously known non-affine family T8^(1) introduced in [3] and the original T8C of [4], so even a perfect search over list (9) cannot establish inequivalence to all previously known families. Section 3.3 explicitly says T8^(1) is 'most likely' a subfamily of a new family, and Proposition 4.5 is supposed to prove containment in (T8D)^T, but its displayed identity is not a valid checkable statement: D(T8D)(e^{i3π/10}u/y, iz, i, −iyz/x, e^{i3π/10}, i, ix) passes seven arguments to T8D, which is defined with six variables. No derivation is supplied, and the notation T8D(T6) is also problematic because T8D is not a Hadamard family on all of T6. The old T8C is dismissed with 'seems to contain' and no proof. These gaps directly affect the novelty conclusion.
[§5.1, Algorithm for determining family membership] Step 1 requires finding all mutually permutation-inequivalent dephased matrices equivalent to H, and Step 3 decides permutation equivalence; both steps depend on the unpublished algorithm [16], and no error analysis is given for the floating-point equality test |z2 − z1| < ε. Consequently the algorithm's 'exhaustive' conclusion is not a mathematical certificate as presented. To make Theorem 5.2 and Proposition 5.3 verifiable, the author should either provide the implementation and exact arithmetic or rigorous interval bounds, or state the theorem as conditional on the correctness of the search.
[§4.4, Lemma 4.9 and Theorem 4.10] The system (7) in Lemma 4.9 lists the fifth and sixth equations identically, both reading Im(ab) Re(ef) − Re(dg) Im(hc) = 0, although they should correspond to the distinct polynomials p_F^(5) and p_F^(6) from Lemma 4.8. As printed, the system has only five independent constraints, and the derivation of the solution in Theorem 4.10 cannot be checked. Also, the formulas for the imaginary parts in Theorem 4.10 contain leading factors of i and a repeated g_i with different expressions, which makes the parameterization internally inconsistent as written.

minor comments (4)

[§4.4, Theorem 4.10] The concluding sentence of Theorem 4.10 says 'T (3)_8C is a 3-parameter family', but the theorem is about T8F; this should be corrected.
[§4.3, Theorem 4.4] The domain is written as D = {(d,e,f) ∈ T^3 | ...}, but the free variables of SD are (a,b,c); the domain should be a subset of T^3 for (a,b,c).
[§5.1, Example 5.1] The notation β = exp(√−3) and γ = exp(√−7) is nonstandard; since these are used as unimodular parameters, the intended expressions exp(i√3) and exp(i√7) should be written explicitly.
[References] Reference [16] is listed as 'in preperation'; this should be 'in preparation', and the manuscript should state more precisely what parts of the equivalence algorithm are assumed versus proved here.

Circularity Check

1 steps flagged · score 6.0 of 10

Inequivalence claim rests on an unpublished self-cited equivalence algorithm; the polynomial construction itself is not circular.

self citation load bearing [Section 5.1, Step 3 and Section 5.2, Theorem 5.2 proof; reference [16].]
"Permutation equivalence of two matrices can be determined with an algorithm from [16]... An Exhaustive computer search with the algorithm described in Section 5.1 yields the result."

The paper's headline claim—that the six families contain matrices not equivalent to any previously known Hadamard matrices—rests on Theorem 5.2, whose proof is only that an exhaustive computer search yields the result. That search's completeness depends on Step 1 (enumerating all dephased equivalents of H) and Step 3 (checking permutation equivalence), and the latter is explicitly delegated to [16], an unpublished manuscript by the same authors. No correctness proof, code, or data is supplied, and [16] is neither machine-checked nor externally reproducible. The negative non-containment result is therefore not derived in this paper; it is assumed from a self-citation, making the central newness claim load-bearing on the authors' own unpublished work.

full rationale

The construction of the new families is self-contained: each family is defined as the image of a function solving the polynomial orthogonality system, and the Hadamard property is enforced by the solved equations rather than assumed. There is no fitted input renamed as a prediction, and the definitions do not presuppose the target result. The circularity concern is confined to the inequivalence/newness claim. Theorem 5.2 and Proposition 5.3 are not mathematical derivations but outputs of a floating-point exhaustive search whose completeness relies on reference [16], an in-preparation paper by the same authors. Under the stated rules this is load-bearing self-citation: the cited algorithm is not machine-checked, code-reproduced, or parameter-free with stated assumptions excluding the target result. Separately, list (9) omits the previously known family T8^(1) and the old T8C^(3), and Proposition 4.5's containment proof is invalid as written (it passes seven arguments to the six-variable function T8D). Those are correctness gaps, not additional circularity, but they reinforce that the newness claim is not independently established. Overall, the derivation of the families is not circular, while the central claim of novelty is supported only by a self-citation chain, giving a partial circularity score of 6.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The construction uses no fitted physical constants. The central claim depends on the correctness of an unreleased computer search, on unpublished equivalence software, and on hand-chosen square-root branches; those are the main uncharged inputs.

free parameters (2)

floating-point tolerance epsilon = unspecified
The membership algorithm in Section 5.1 declares complex numbers equal when |z2-z1| < epsilon; the choice of epsilon affects whether equivalences are detected and hence the conclusion that families are new. No value is given.
square-root branch choices in d', e', f' = multiple sign choices; numerical results suggest equivalence
The domains in Theorems 4.2, 4.4, and 4.10 require choosing branches of square roots. The paper states that other sign branches give equivalent families based on numerical evidence, not proof, so the chosen branch is a hand-made choice the construction depends on.

assumptions (4)

domain assumption The exhaustive backtracking search over all permutation-inequivalent dephased equivalents of a matrix terminates and finds all equivalences.
Theorem 5.2 and Proposition 5.3 depend on this search being complete; the search is described but not proven and relies on the unpublished equivalence algorithm [16].
domain assumption Floating-point arithmetic with tolerance epsilon correctly decides exact complex-algebraic equalities.
Section 5.1 states that complex numbers are considered equal if |z2-z1| < epsilon; the search conclusions, including non-equivalence, inherit this approximation.
domain assumption The evaluation points (e^{2i}, ..., e^{(k+1)i}) belong to the domains of all listed families.
The proof of Theorem 5.2 asserts 'One can check' without showing that the square-root branch domains such as C/(a^8 b^8 c^4) <= 0 are satisfied.
standard math Standard algebraic manipulation of palindromic polynomials is valid.
Lemmas 2.4 through 2.6 are elementary and appear correct; this is background math used throughout the construction.

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Cite this review

Pith. "Pith review of Non-affine Families of 8 x 8 Complex Hadamard Matrices." pith.science (2026). https://pith.science/paper/LZFGOBJZ

@misc{pith2026250522947,
  author       = {Pith},
  title        = {Pith review of: Non-affine Families of 8 x 8 Complex Hadamard Matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LZFGOBJZ}},
  note         = {Machine review of arXiv:2505.22947}
}

read the original abstract

Six non-affine 3-parameter families of complex Hadamard matrices of order 8 are presented. These families contain Hadamard matrices that are not equivalent to any previously known Hadamard matrices in the literature. Each family arises from unimodular points of an affine variety defined by palindromic polynomials. The families are given as an image of a function that solves the corresponding system of polynomials on a domain that guarantees unimodularity of the solutions

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [4]

W. Bruzda. Block-circulant complex Hadamard matrices. J. Math. Phys., 64:Paper No. 052201, 17, 2023

work page 2023

[2] [16]

Musto and J

B. Musto and J. Vicary. Quantum Latin squares and unitary error bases. Quantum Inf. Comput., 16:1318–1332, 2016

work page 2016

[3] [3]

W. Bruzda. Extension of the set of complex Hadamard matrices of size

work page

[4] [1]

S. S. Agaian. Hadamard Matrices and Their Applications. Springer, Heidelberg, 1985

work page 1985

[5] [2]

Bengtsson, W

I. Bengtsson, W. Bruzda, Å. Ericsson, J.-Å. Larsson, W. Tadej, and K. Życzkowski. Mutually unbiased bases and Hadamard matrices of order six. J. Math. Phys., 48, 2007

work page 2007

[6] [5]

Bruzda, W

W. Bruzda, W. Tadej, and K. Życzkowski. Complex Hadamard matrices - a Catalog. 2006. https://chaos.if.uj.edu.pl/~karol/hadamard/ index.html, Accessed: 2024-09-25

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P. Dita. Hadamard matrices from mutually unbiased bases.J. Math. Phys., 51:072202, 20, 2010. 22

work page 2010

[8] [7]

Durt, B.-G

T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski. On mutually unbiased bases.Int. J. Quantum Inf., 8:535–640, 2010

work page 2010