arxiv: 2605.08661 · v1 · submitted 2026-05-09 · 🧮 math.CO

Recognition: no theorem link

A search for Hadamard matrices of Williamson type

Ali Mohammadian, Behruz Tayfeh-Rezaie, Hadi Kharaghani

Pith reviewed 2026-05-12 00:58 UTC · model grok-4.3

classification 🧮 math.CO

keywords Hadamard matricesWilliamson matricesnear Williamson matricesquaternary Hadamard matricescirculant matricescomputer searchcombinatorial constructionsorthogonal matrices

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The pith

Near Williamson matrices exist for all odd orders up to 63, giving the first quaternary Hadamard matrix of order 118.

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

The authors define near Williamson matrices as four n by n matrices with entries plus or minus one, where one is circulant and the other three are symmetric circulant, and they satisfy the sum of their Gram matrices equaling 4n times the identity. They run an exhaustive computer search to list all inequivalent examples for every odd order at most 35. They also supply explicit constructions proving existence for every odd order through 63. This work directly produces a quaternary Hadamard matrix at order 118, which had not been known before. Readers interested in combinatorial designs would care because these matrices generate new orthogonal arrays and codes with potential use in communications and experimental design.

Core claim

We introduce near Williamson matrices: four n by n (-1,1)-matrices A, B, C, D such that A is circulant, B, C and D are symmetric circulant, and AA transpose plus BB transpose plus CC transpose plus DD transpose equals 4n times the identity matrix. Computer enumeration finds all inequivalent quadruples for every odd n at most 35. Explicit constructions are given for every odd n up to 63. As a direct consequence, a quaternary Hadamard matrix of order 118 is obtained.

What carries the argument

Near Williamson matrices: four specially circulant (-1,1)-matrices whose Gram-matrix sum equals 4n times the identity.

If this is right

All inequivalent near Williamson matrices for odd orders at most 35 are now known.
Near Williamson matrices exist for every odd order up to 63.
A quaternary Hadamard matrix of order 118 exists and can be written down explicitly.
The same construction method produces new Hadamard matrices from near Williamson quadruples at other orders.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The classification data for small orders supplies test cases for any future theoretical classification of Williamson-type matrices.
If near Williamson matrices can be built for all odd orders, the same route would generate quaternary Hadamard matrices at many additional orders.
The order-118 example can be checked directly by matrix multiplication to confirm it meets the Hadamard orthogonality condition.

Load-bearing premise

The computer search for orders up to 35 is exhaustive and free of implementation errors, and the explicit constructions up to order 63 are correctly verified.

What would settle it

An independent check that fails to locate any near Williamson matrix for some odd order at most 35, or that shows the constructed order-118 matrix does not satisfy the quaternary Hadamard condition, would disprove the results.

read the original abstract

In this article, we consider a special class of Williamson type matrices which we call them near Williamson matrices. They are in fact four $n\times n$ $(-1, 1)$-matrices $A, B, C, D$ so that $A$ is circulant, $B,C,D$ are symmetric circulant, and they satisfy $AA^\top+BB^\top+CC^\top+DD^\top=4nI$. Using a computer search, we find all inequivalent near Williamson matrices for all odd orders at most $35$. We also show that such matrices exist for all odd orders up to $63$. As a consequence, we find the first known example of a quaternary Hadamard matrix of order $118$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Computer enumeration of near Williamson matrices up to 35 gives a new Hadamard matrix at 118, but the search lacks enough detail for easy verification.

read the letter

The main thing here is a computational enumeration of near Williamson matrices for odd orders up to 35, plus constructions to 63 that yield the first quaternary Hadamard matrix of order 118. The authors define these as four n by n matrices with ±1 entries where one is circulant and the other three are symmetric circulant, satisfying the standard sum of transposes times themselves equals 4n times the identity. They use a computer search to find all inequivalent versions for odd n at most 35 and give explicit constructions for existence up to 63. The 118 order matrix follows from that. This adds a new concrete example to the Hadamard matrix literature, which is always a plus in this field. The small-order enumeration also provides a useful reference for anyone checking patterns or trying to extend results. The soft spot is the reliance on the computer search without enough supporting material. No code is provided, and the details on how they managed the search space, tested for inequivalence, or handled the symmetric cases are not thorough enough to allow straightforward independent verification. If there's an error in the enumeration at some order like 35 or 59, it would impact the novelty of the 118 matrix. That's the main limitation. This paper is for researchers in combinatorial design theory who focus on Hadamard matrices and Williamson-type constructions. A reader looking for new examples or data on small cases will find it relevant. It deserves to go through peer review so that experts can assess the computational claims and the constructions in detail.

Referee Report

3 major / 3 minor

Summary. The paper defines near Williamson matrices as four n×n (-1,1)-matrices A (circulant), B,C,D (symmetric circulant) satisfying AAᵀ + BBᵀ + CCᵀ + DDᵀ = 4nI. It reports a computer search that enumerates all inequivalent examples for every odd order n ≤ 35, gives explicit constructions establishing existence for all odd n ≤ 63, and uses the n=59 case to produce the first known quaternary Hadamard matrix of order 118.

Significance. If the enumeration and constructions hold, the work supplies a complete classification for small odd orders and extends the existence range to 63, with the order-118 example constituting a concrete new construction in the Hadamard-matrix literature. The explicit constructions themselves are a verifiable strength that supports further use.

major comments (3)

[Computer-search section] Section describing the computer search: the enumeration algorithm is not specified in sufficient detail (backtracking order, isomorphism test under the dihedral action on circulants, symmetry reductions for the symmetric-circulant cases). No source code, raw output files, or machine-readable lists of the matrices found for n=35 are supplied, so the claim that all inequivalent near Williamson matrices were found for every odd n≤35 cannot be independently verified.
[Constructions section] Section on constructions for orders up to 63: the explicit construction for n=59 (the case that yields the claimed quaternary Hadamard matrix of order 118) is not exhibited with matrix entries or a direct verification that the four matrices satisfy the defining orthogonality relation; only the existence statement is given.
[Enumeration results] Table or list of enumerated matrices (if present): without the actual matrices or a checksum for the n=35 case, the completeness assertion for the search up to 35 remains uncheckable and therefore load-bearing for any claim that relies on the classification.

minor comments (3)

[Abstract] The abstract introduces the term 'quaternary Hadamard matrix' without a brief definition or forward reference to the precise construction used.
[Definitions] Notation for the four matrices A,B,C,D and the precise meaning of 'inequivalent' under the relevant group action should be stated once in the definitions section and used consistently thereafter.
[Application to Hadamard matrices] A short paragraph or appendix reference explaining how the n=59 near Williamson matrices produce a quaternary Hadamard matrix of order 118 would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We agree that additional details are needed to make the computational results fully verifiable, and we will revise the manuscript accordingly.

read point-by-point responses

Referee: [Computer-search section] Section describing the computer search: the enumeration algorithm is not specified in sufficient detail (backtracking order, isomorphism test under the dihedral action on circulants, symmetry reductions for the symmetric-circulant cases). No source code, raw output files, or machine-readable lists of the matrices found for n=35 are supplied, so the claim that all inequivalent near Williamson matrices were found for every odd n≤35 cannot be independently verified.

Authors: We acknowledge that the description of the search algorithm in the manuscript is insufficient for independent verification. In the revised version, we will provide a detailed description of the backtracking procedure, including the order in which variables are assigned, the method for testing isomorphism under the dihedral group action, and the symmetry reductions applied to the symmetric circulant matrices. Additionally, we will make the source code available via a public repository and supply machine-readable files containing the complete lists of inequivalent near Williamson matrices for all odd n up to 35. revision: yes
Referee: [Constructions section] Section on constructions for orders up to 63: the explicit construction for n=59 (the case that yields the claimed quaternary Hadamard matrix of order 118) is not exhibited with matrix entries or a direct verification that the four matrices satisfy the defining orthogonality relation; only the existence statement is given.

Authors: We agree that exhibiting the explicit matrices for n=59 would strengthen the paper. In the revision, we will include the four matrices A, B, C, D for n=59, along with a brief verification or checksum confirming that they satisfy AAᵀ + BBᵀ + CCᵀ + DDᵀ = 4nI. revision: yes
Referee: [Enumeration results] Table or list of enumerated matrices (if present): without the actual matrices or a checksum for the n=35 case, the completeness assertion for the search up to 35 remains uncheckable and therefore load-bearing for any claim that relies on the classification.

Authors: To address this, we will augment the enumeration results section with a checksum (such as a hash of the concatenated matrix entries) for the n=35 case, and provide the full set of matrices in a supplementary file or online repository to allow verification of the completeness claim. revision: yes

Circularity Check

0 steps flagged

No circularity; results from direct search and explicit constructions

full rationale

The paper reports exhaustive computer enumeration of near Williamson matrices for all odd orders ≤35 together with explicit constructions up to order 63, from which a quaternary Hadamard matrix of order 118 is obtained as a direct consequence. No derivation chain, fitted parameter, self-definitional equation, or load-bearing self-citation is present; the claimed results are outputs of an independent computational procedure and constructive existence proofs rather than any quantity that reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper introduces the new class of near Williamson matrices but relies on standard matrix algebra over the reals and computational enumeration rather than new free parameters or unverified entities.

axioms (2)

domain assumption The condition AA^T + BB^T + CC^T + DD^T = 4n I is the defining orthogonality relation for Williamson-type matrices
Invoked directly in the definition of near Williamson matrices and the link to Hadamard matrices.
standard math Circulant and symmetric circulant matrices admit the algebraic structure used for enumeration
Used throughout the computer search and existence arguments.

invented entities (1)

near Williamson matrices no independent evidence
purpose: A relaxed subclass of Williamson matrices to enable exhaustive search and constructions
New definition introduced in the paper to facilitate the reported results.

pith-pipeline@v0.9.0 · 5428 in / 1330 out tokens · 66635 ms · 2026-05-12T00:58:42.023527+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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