[#P22] Hadamard matrix conjecture

Problem. For every positive integer \(k\), there exists a matrix \(H\in\{\pm1\}^{4k\times4k}\) such that \(HH^{\mathsf T}=4kI_{4k}\).

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Definitions and notation

1Context

Orders 1 and 2 occur, and every larger possible order must be divisible by 4. The conjecture says this necessary divisibility condition is sufficient.

2Problem setup

Definition 1 (A Hadamard matrix H of order n has entries in {+1,-1} and satisfies H H^T = n I). A Hadamard matrix H of order n has entries in {+1,-1} and satisfies H H^T = n I.

Definition 2 (Pairwise orthogonal rows have dot product zero). Pairwise orthogonal rows have dot product zero.

Remark 1. Orders 1 and 2 occur, and every larger possible order must be divisible by 4. The conjecture says this necessary divisibility condition is sufficient.

3What counts as a solution

Construct a Hadamard matrix of order 4k for every positive integer k, or find a positive integer k and prove that no such matrix exists.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Cati and Pasechnik provide reproducible constructions through order 1208 for all known cases in that range and current tables beyond it. Their database improves many entries but continues to state the universal order-4k assertion as a conjecture. Exact unresolved remainder: Construct a Hadamard matrix of order 4k for every positive integer k, or give a positive k and prove that no matrix of order 4k exists.^[1][2]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Cati and Pasechnik provide reproducible constructions through order 1208 for all known cases in that range and current tables beyond it. Their database improves many entries but continues to state the universal order-4k assertion as a conjecture. Exact unresolved remainder: Construct a Hadamard matrix of order 4k for every positive integer k, or give a positive k and prove that no matrix of order 4k exists.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Cati and Pasechnik provide reproducible constructions through order 1208 for all known cases in that range and current tables beyond it. Their database improves many entries but continues to state the universal order-4k assertion as a conjecture. Exact unresolved remainder: Construct a Hadamard matrix of order 4k for every positive integer k, or give a positive k and prove that no matrix of order 4k exists.^[1][2] Relevance to this problem For Hadamard matrix conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Cati and Pasechnik provide reproducible constructions through order 1208 for all known cases in that range and current tables beyond it. Their database improves many entries but continues to state the universal order-4k assertion as a conjecture. Exact unresolved remainder: Construct a Hadamard matrix of order 4k for every positive integer k, or give a positive k and prove that no matrix of order 4k exists. Open full record →
Construct a Hadamard matrix of order 4k for every positive integer k, or find a positive integer k and prove that no such matrix exists.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Construct a Hadamard matrix of order 4k for every positive integer k, or find a positive integer k and prove that no such matrix exists.^[1] Relevance to this problem For Hadamard matrix conjecture, record hadamard-matrix-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Construct a Hadamard matrix of order 4k for every positive integer k, or find a positive integer k and prove that no such matrix exists. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Construct a Hadamard matrix of order 4k for every positive integer k, or find a positive integer k and prove that no such matrix exists. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited 2024 research paper presents the Hadamard existence statement as a conjecture. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Many infinite families and individual orders are known. Check current construction tables before claiming a new order.

Recorded example 1. The 4 by 4 matrix with first row all +1 and subsequent rows (+1,-1,+1,-1), (+1,+1,-1,-1), and (+1,-1,-1,+1) is Hadamard.

Computational notes

Computer searches can settle particular orders and test construction methods.

2See also

Nonvanishing of Rudin-Shapiro Hankel determinantslinear algebra
Nonvanishing of Baum-Sweet Hankel determinantslinear algebra
Ternary representatives of finite-order integral matriceslinear algebra

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Plain text

“Hadamard matrix conjecture.” TheoremDB. P22. Problem statement; statement text SHA-256 bd7c632555d860630389eec4731418fc5eccdcb4abce923d7b1d0be847d380b9. https://theoremdb.org/statement/?ref=P22

BibTeX

@misc{theoremdb-problem-bd7c632555d860630389eec4731418fc5eccdcb4abce923d7b1d0be847d380b9,
  title = {{Hadamard matrix conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 bd7c632555d860630389eec4731418fc5eccdcb4abce923d7b1d0be847d380b9},
  url = {https://theoremdb.org/statement/?ref=P22}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Stefan Steinerberger, “A Note on Approximate Hadamard Matrices”. arXiv:2402.13202 (2024). Stefan Steinerberger, arXiv:2402.13202, abstract and section 1.1. ↗preprint · primary source · arXiv:2402.13202, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited 2024 research paper presents the Hadamard existence statement as a conjecture. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at abstract and section 1.1.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Provides nearby approximate-matrix results and states the exact Hadamard existence target.Source named by the research packet.
Matteo Cati and Dmitrii V. Pasechnik, “A database of constructions of Hadamard matrices”. arXiv:2411.18897 (2024). abstract, SageMath construction range, and current tables. ↗preprint · primary source · arXiv:2411.18897v2 · checked 2026-08-01Source use: original summary.Provides the strongest checked reproducible construction database while retaining the universal conjecture.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P22] Hadamard matrix conjecture

1Context

2Problem setup

3What counts as a solution

1Status

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Computational notes

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1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

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Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance