[#P22] Hadamard matrix conjecture
Problem. For every positive integer \(k\), there exists a matrix \(H\in\{\pm1\}^{4k\times4k}\) such that \(HH^{\mathsf T}=4kI_{4k}\).
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
Current 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]
1Records
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
- Ternary representatives of finite-order integral matriceslinear algebra
- Orders of ternary row-orthogonal matrices with a full rowlinear algebra
- Rota's basis conjecturelinear algebra
How to cite
TheoremDB contributors, “Hadamard matrix conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/hadamard-matrix-conjectureThis page as plain text: hadamard-matrix-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
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.