Construction of Butson matrices using Fourier matrices as input

Farouk Adda

REVIEW 2 major objections 5 minor 16 references

Construction of Butson matrices using Fourier matrices as input

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper gives block constructions that turn a Butson matrix of order $n$ into Butson matrices of orders $n(n-1)$ and, under extra conditions, $n(n/2-1)$, using Latin-square arrangements called LSESC to control orthogonality.

desk verdict The conditional construction is real and the proofs hold, but the paper's universal claim rests on a self-cited existence theorem that contradicts known MOLS bounds. read the letter →

arxiv 2504.15980 v1 pith:NPYKHRK2 submitted 2025-04-22 math.CO

Farouk Adda This is my paper · ORCID

classification math.CO MSC 15B2005C31

keywords ButsonmatricescomplexHadamardScarpisconstructionLatinsquaresmutuallyorthogonalLSESCFourier

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 proposes two families of constructions that produce larger Butson matrices from smaller ones. Theorem 1 turns any normalized Butson matrix of order $n$ into a Butson matrix of order $n(n-1)$, and Theorem 2 turns any Butson matrix in a specified subset $A(1,2)(m,n)$ into one of order $n(n/2-1)$, with $n$ and $m$ even in the second case. Both constructions come in one-input and two-input versions, so the procedures can produce many matrices of the same order. The output entries remain $m$-th roots of unity, which makes this a systematic way to enlarge the supply of complex Hadamard-style orthogonal matrices, including ordinary Hadamard matrices. The author also counts the outputs and shows that Fourier matrices can serve as inputs.

What carries the argument

The load-bearing objects are LSESC, Latin squares eligible for Scarpis construction: two Latin squares $L,L'$ of order $q$ such that for every pair $(i,i')$ there is a unique $j$ with $L_{ij}=L'_{i'j}$. A complete set has $q-1$ pairwise eligible squares, and the paper notes that such sets are conjugate to mutually orthogonal Latin squares, so the maximal number $N'(q)$ equals $N(q)$. Each eligible square is encoded as a cubic tensor whose frontal slices are permutation matrices, and the tensor set is woven into the blocked matrix of (3.1) or (4.5). The eligibility condition is what makes cross-block row inner products collapse to zero, because it guarantees exactly one collision between the Latin-square row entries in each cross term.

What would settle it

Run the claimed algorithm from [5] for order 6 and check whether it really produces a complete set of LSESC of order 6; because the paper equates LSESC sets with mutually orthogonal Latin squares, the known nonexistence of a complete set of MOLS of order 6 would already predict failure, and this would limit the constructions to orders where complete LSESC sets exist.

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Extended reading notes

Core claim

The central claim is that a Scarpis-style block matrix built from an input Butson matrix and a complete tensor set of LSESC is again a Butson matrix. Theorem 1 states that for $H \in BH(m,n)$ with core $C$ and any complete tensor set $X$ of LSESC of order $n-1$, the matrix $\Phi_X(H)$ defined in (3.1) lies in $BH(m,(n-1)n)$. Theorem 2 states that for $H \in A(1,2)(m,n)$ with $n$ even and a complete tensor set of LSESC of order $n/2-1$, the matrix $\Psi_X(H)$ defined in (4.5) lies in $BH(m,(n/2-1)n)$. Orthogonality is proved by checking row orthogonality in four cases, with column orthogonality following from the same structure. The paper also gives two-input variants $\Phi_X(G,H)$ and $\Psi_X(G,H)$, applies the machinery to Hadamard matrices, and identifies conditions under which Fourier matrices can serve as the input.

Load-bearing premise

The construction needs a full set of specially arranged Latin squares of size $n-1$ (or $n/2-1$) for every relevant $n$, and the paper takes the existence of such sets from the author's earlier work; yet the same paper identifies these arrangements with mutually orthogonal Latin squares, and no complete set of those exists for order 6.

Editorial extensions

If this is right

For every $n$ for which a complete set of LSESC of order $n-1$ exists, every $BH(m,n)$ produces a $BH(m,n(n-1))$, and the two-input version produces $|MOLS(n-1)|\cdot|BH(m,n)|^2\cdot n$ candidate matrices.
For even $n$ with a complete set of LSESC of order $n/2-1$, every input from $A(1,2)(m,n)$ produces a $BH(m,n(n/2-1))$, and the two-input version widens the eligible first matrix to all of $A(1)(m,n)$.
Applied to Hadamard matrices, the same block templates produce Hadamard matrices of orders $n(n-1)$ and $n(n/2-1)$, extending the author's earlier Scarpis-type Hadamard constructions.
Fourier matrices satisfy the needed conditions in the relevant cases, so the constructions give explicit new matrices, such as the $BH(6,12)$ example built from $F_6$.

Reading between the lines

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

Editorial inference: because LSESC sets are identified with MOLS, the claim that complete LSESC sets exist for every order sits in tension with the classical fact that a complete set of MOLS of order 6 does not exist; if that tension is real, the constructions apply only where complete sets genuinely exist rather than for all $n$.
Editorial inference: the two-input variants separate the role of the first matrix (which supplies row-sign structure) from the second (which supplies the core), so inequivalent input pairs should often produce inequivalent outputs; the counting formulas then bound new classes only from below, and the paper leaves this classification open.
A testable extension would feed a fixed input matrix with all known complete LSESC sets of small order and compare normalized outputs against existing classifications of complex Hadamard matrices to count genuinely inequivalent outputs.

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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

Dear [Colleague],

You should know this paper has a real conditional construction hidden behind an overblown universal claim. Theorems 1 and 2 give explicit block matrices that are BH(m,(n-1)n) and BH(m,(n/2-1)n) whenever you have a complete set of LSESC of the appropriate order and, for Theorem 2, an input matrix in A(1,2)(m,n). The row-orthogonality proofs are straightforward and, as far as I can tell, correct. I checked the four cases in Theorem 1 and the analogous ones in Theorem 2; the use of the LSESC property in the cross-block case is exactly what you'd need. The two-input variants in Corollaries 1 and 4 are a genuine extension of [8], which only handles the single-input, prime-power case. The tensor formalism for LSESC is a little heavy but it works.

The soft spot is the existence problem. The paper claims the construction applies to all n because [5, Theorem 2.6] allegedly produces n-1 LSESC of every order n. But Section 2 itself says LSESC are conjugate to MOLS and that a complete set of MOLS of order 6 does not exist (the conjecture is verified through order 6). If Theorem 2.6 were true for n=6, it would give 5 LSESC, hence 5 MOLS of order 6, contradicting N(6)=1. That is an internal inconsistency in the manuscript, not just a gap. The author needs to either restrict the theorems to orders for which complete LSESC sets exist or provide a correct existence proof that is compatible with MOLS bounds. As a consequence, Corollary 3 is false for r=3 (n=14), where the construction would require a complete LSESC set of order 6. The order formula in Corollary 3 also appears wrong: for n=2(2r+1), the construction gives order 4r(2r+1), not the displayed value.

The paper deserves a serious referee because the conditional construction is novel and likely correct, and the flaws are fixable in revision. But it is not acceptable as-is. A referee should ask for the existence claim to be substantiated or weakened, and for the corollaries to be rechecked.

In short: worth engaging, but the advertised scope currently overreaches.

Referee Report

2 major / 5 minor

Summary. The manuscript proposes two conditional constructions of Butson matrices. Theorem 1 constructs a BH(m,(n-1)n) matrix from a BH(m,n) matrix, given a complete tensor set of LSESC of order n-1. Theorem 2 constructs a BH(m,(n/2-1)n) matrix from a BH(m,n) matrix in A(1,2)(m,n), given a complete tensor set of LSESC of order n/2-1. Two-input versions and applications to Hadamard matrices are also given. The proofs verify row orthogonality directly. The paper further claims that these constructions apply for every n because the author's earlier work [5] supplies complete sets of LSESC of every order, and it states Corollary 3 asserting the existence of certain matrices from Fourier inputs for all r.

Significance. Conditional on the existence of the required LSESC inputs, the constructions appear algebraically sound and provide a new, potentially useful method for enlarging Butson orders, with explicit counts of output matrices. The Hadamard variants and the use of Fourier matrices as inputs are of interest. However, the paper's universal claims rest on a self-cited existence result that is internally inconsistent with the known non-existence of complete MOLS of order 6, and the proofs of Theorem 2 contain an unstated assumption about the entries of the C1 row. These issues materially affect the advertised scope.

major comments (2)

[Section 2 and Section 4, Corollary 3] The unconditional applicability of Theorems 1 and 2 is claimed on the basis of the self-cited Theorem 2.6 of [5], which is said to construct a complete set of LSESC of size n-1 for every n. This contradicts the facts stated earlier in the same section: the paper asserts N'(n)=N(n) and that the complete-MOLS conjecture is verified for orders at most 6, so a complete set of 5 LSESC of order 6 would conjugate to a complete set of 5 MOLS of order 6, contradicting N(6)=1. Consequently, Theorem 2 with n=14 requires a complete tensor LSESC set of order 6, which cannot exist under the paper's own assumptions, and Corollary 3 cannot hold for all r as stated. The authors should either supply a valid existence proof for the required LSESC sets, or explicitly restrict Theorems 1-2 and Corollary 3 to orders for which complete LSESC sets are known to exist (e.g., prime powers), and remove the conflicting claims about [5].
[Theorem 2, case 3, Eq. (4.10)] The proof of Theorem 2 uses 'xlxl = 1 for all l' to cancel the factors x_l^2 in the inner products of rows of Gamma_k. The row (x1 ... xn) is only required by condition C1 to be a row of H with the alternating-sign row also present; its entries are arbitrary m-th roots of unity and are not necessarily +-1. Unless the construction explicitly chooses (x1 ... xn) to be the all-ones row or the {1,-1} row from condition C2, the factorization x_l^2=1 is unjustified and the orthogonality argument for case 3 does not go through. Please state this choice, prove it, or restrict Theorem 2 to a subclass where it holds.

minor comments (5)

[Abstract] The abstract states that both constructions work when n and m are even, but Theorem 1 does not require m even; only Section 4's construction uses the evenness through Lemma 1. Please correct or qualify the abstract.
[Example 1] The sentence 'j^2 = j' is a typo; for j = e^{2*pi*i/3}, j^2 = j^{-1}, not j. Please correct.
[Corollary 3] The expression 'BH(2(2r + 1), 2r+1(2r + 1))' appears to have a missing superscript and does not match the output order n(n/2-1) = 4r(2r+1) when n=2(2r+1). Please correct the formula and clarify the notation.
[Section 3, counting paragraph] The count '|MOLS(n-1)| * |BH(m,n)|^2 * n' is stated without proof; since the constructions depend on complete tensor LSESC sets, the count should be justified in terms of the number of such sets and the choices of rows and columns.
[Throughout] There are numerous typos and typesetting errors, including 'dimenssions' in the Introduction, 'imputed' instead of 'input' in Section 4, and garbled matrix displays in Lemma 2. A thorough proofread is needed.

Circularity Check

1 steps flagged · score 4.0 of 10

Main construction proofs are direct and self-contained; only the universal LSESC-existence premise is load-bearing self-citation, and it conflicts with the paper's own N'(n)=N(n) statement at order 6.

self citation load bearing [Section 2, LSESC/MOLS discussion; assumed in Theorems 1–2 and Corollary 3]
"However, we can also construct a complete set of LSESC that is not a set of MOLS, as demonstrated in Theorem 2.6 in [5]. This theorem proposes an algorithm for constructing a set of LSESCs of size n − 1 of each order n ... By taking a set of LSESC and, for each entry l i j = a, exchanging a and i, the result is a set of MOLS, and vice versa. These two sets are conjugate, and the notions are equivalent. Thus, it follows that N ′(n) = N (n)."

Theorems 1 and 2 are conditional on the existence of complete tensor sets of LSESC of the required orders, but the paper's universal reading, and specifically Corollary 3, depends on such sets existing for every needed order. The only support offered for this existence is Theorem 2.6 of the author's own previous paper [5]. That self-citation is load-bearing, not merely incidental. Moreover, the same section proves that LSESC sets and MOLS are conjugate with N'(n)=N(n), and reports that the MOLS conjecture has been verified for orders at most 6, meaning no complete set of 5 MOLS of order 6 exists. A complete LSESC set of order 6 would therefore contradict the paper's own N'(n)=N(n) statement.

full rationale

The two construction theorems are not circular in their main derivations. Given an input BH(m,n), a core C, and a complete tensor set of LSESC, the proofs verify row and column orthogonality block-by-block using only the defining property of LSESC, the orthogonality and core-sum properties of the input Butson matrix, and the sign structure established in Lemma 2. No parameter is fitted, no output is identified with an input by definition, and the worked examples with F3 and F6 are independently checkable. The circularity burden is confined to the existence of the complete LSESC sets themselves. The paper does not construct these sets here; it imports their existence from Theorem 2.6 of [5], a self-citation. That imported premise is load-bearing for the advertised universal scope and for Corollary 3, and it is internally inconsistent with the paper's own statement that LSESC and MOLS are conjugate with N'(n)=N(n), since complete LSESC of order 6 would imply complete MOLS of order 6, contrary to the verified conjecture for orders up to 6. Thus the conditional proofs are sound and self-contained, but the universal existence claim rests on a non-independent, self-cited premise that the paper itself undermines.

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

The central derivation is conditional on given LSESC inputs; no free parameters are fit. The load-bearing assumption is the availability of complete LSESC sets, imported from the author's earlier paper [5].

assumptions (4)

domain assumption Complete sets of LSESC of order q exist for the q values used (q = n-1 and q = n/2-1).
Theorem 1 requires |X| = n-2 LSESC of order n-1; Theorem 2 requires |X| = n/2-2 of order n/2-1. The paper cites [5] for existence for every order, but does not prove it here; because LSESC are claimed equivalent to MOLS, universal existence would conflict with known bounds for order 6.
standard math Normalized Butson matrices have a core with row sums -1 and pairwise row products -1.
Used throughout the proofs, in particular Theorem 1 relies on core row sums and -1 dot products; this follows from normalization and orthogonality.
standard math The Kronecker product preserves orthogonality of rows when the second factor has orthogonal rows.
Used in Theorems 1 and 2 for B0 = H' tensor J, where J is the all-ones row vector.
standard math The column property of Latin squares and the pairwise LSESC property give exactly one matching symbol in cross-block row pairs.
Used in the final case of the Theorem 1 proof and in the analogous cross-block cases of Theorem 2.

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Cite this review

Pith. "Pith review of Construction of Butson matrices using Fourier matrices as input." pith.science (2026). https://pith.science/paper/NPYKHRK2

@misc{pith2026250415980,
  author       = {Pith},
  title        = {Pith review of: Construction of Butson matrices using Fourier matrices as input},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NPYKHRK2}},
  note         = {Machine review of arXiv:2504.15980}
}

abstract

Butson matrices are square orthogonal matrices, denoted by $BH(m,n)$, whose entries are the complex $m$th roots of unity and satisfy the condition\\ $BH(m,n)\cdot{BH(m,n)}^*=nI_n$, where ${BH(m,n)}^*$ is the conjugate transpose of $BH(m,n)$ and $I_n$ is the identity matrix. In this work, we propose constructions for $BH(m,(n-1)n)$ then $BH(m,(\frac{n}{2}-1)n)$, when $n$ and $m$ are even numbers, using the existing $BH(m,n)$. For each case, we provide two construction methods: one uses a single input Butson matrix, and another uses two input Butson matrices. Moreover, we present some results about the construction of Hadamard matrices.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [8]

Sargent, K

M. Sargent, K. Lee, J. Rushall, On Generalizing the Metho d of Scarpis to Complex Hadamard Matrices, Journal of Combinatorial Mathe matics and Combinatorial Computing, 119: 105–111

work page

[2] [5]

Farouk, Q

A. Farouk, Q. W. Wang, On the construction of Hadamard mat rices by using the known Hadamard matrices, FILOMAT, (2022) 36(6): 2 025–2042

work page 2022

[3] [1]

K. J. Horadam, Hadamard matrices and their applications , Princeton Uni- versity Press, Princeton. NJ, 2007

work page 2007

[4] [2]

A. B. Evans, Orthogonal Latin Squares Based on Groups, Sp ringer Inter- national Publishing AG, part of Springer Nature, 2018

work page 2018

[5] [3]

T. G. Kolda, B. W. Bader, Tensor Decompositions and appli cations, SIAM REVIEW, (2009) 51(3): 455–500

work page 2009

[6] [4]

Farouk, Q

A. Farouk, Q. W. Wang, An inﬁnite family of Hadamard matri ces con- structed from Paley type matrice, FILOMAT, (2020) 34(3): 81 5–834

work page 2020

[7] [6]

Scarpis, Sui determinanti di valore massimo [On deter minants of maxi- mal value], Rendic R Istit Lombardo Sci Lett, (1898) 31: 1441 –1446

U. Scarpis, Sui determinanti di valore massimo [On deter minants of maxi- mal value], Rendic R Istit Lombardo Sci Lett, (1898) 31: 1441 –1446

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Dokovi` c, Generalization of Scarpis’ theorem on Hadamar d matrices, Linear and Multilinear Algebra, (2017) 65: 1985–1987

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