# P2520: A Hadamard matrix of order 668

- ID: `P2520`
- Reference: `hadamard-order-668`
- Page: https://theoremdb.org/statements/P2520
- Record maturity: Reviewed problem with recorded work

## Problem

Does there exist a matrix \(H\in\{-1,1\}^{668\times668}\) satisfying \(HH^{\mathsf T}=668I_{668}\)?

### Remarks

- **Remark.** The matrix equation says that every two distinct rows have dot product zero.
- **Remark.** Negating rows or columns and permuting rows or columns preserve the Hadamard property.

### What counts as a solution

- Supply the 668 rows of \(H\) and an exact integer check of \(HH^{\mathsf T}=668I_{668}\).

## Status

No real Hadamard matrix of order 668 is known, and no nonexistence proof is known. [3](#reference-3) [2](#reference-2) [4](#reference-4) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Order 668 remains open).** No real Hadamard matrix of order 668 is known, and no nonexistence proof is known.

The current answer to the candidate is open. Cati and Pasechnik's construction database, revised on August 30, 2025, states that all known Hadamard orders through 1208 are implemented in SageMath and lists 668, 716, 892, and 1132 as the unknown orders in that range. Their Section 3 gives the direct query
```
hadamard_matrix(668, existence=True)
Unknown
```
The result `Unknown` means that SageMath has no construction and makes no nonexistence assertion.

The appendix gives a second check. Table 2 lists a construction for each known order \(4n\) with odd \(n<300\); the entry at \(n=167\) is blank. Table 4 records \(167(3)\), meaning that the least known exponent \(m\) for an order \(2^m167\) construction is \(m=3\). Thus order \(1336=8\cdot167\) is known, while the desired order \(668=4\cdot167\) is the missing exponent-two case.

Eliahou's 2025 primary paper calls 668 the smallest open case and constructs the 64-modular near miss checked in the companion artifact. Suksmono's 2025 search paper independently lists 668, 716, and 892 as unresolved below 1000. Epoch AI still identifies 668 as the smallest unknown order on the access date. These sources report the state of knowledge. They cannot prove that a matrix does not exist.

### Background and intake notes

The order factors as \(668=4\cdot167\). A witness would settle the first unresolved case of the Hadamard conjecture and can be checked with integer arithmetic.

- Original intake status: As of 2026-07-24, 668 is reported as the smallest positive multiple of four for which no Hadamard matrix is known.
- Normalize the first row and column to all ones before comparing searches. Store the construction family, parameter restrictions, and exact residual autocorrelations for every structured attempt.
- Goethals-Seidel arrays, Williamson-type sequences, and plug-in block constructions reduce the matrix equation to smaller exact constraints. Exhausting one family leaves the other families open.

### Other known results

- **Computation 1** (reproduced): Eliahou's matrix has Gram matrix congruent to 668I modulo 64, while every row has 26 nonzero off-diagonal dot products over the integers. [2](#reference-2)

### Prior approaches

- **Route 1** (supported): Current tables and recent primary papers agree that 668 is unresolved; the strongest explicit order-668 object found is modular. [3](#reference-3)

### Runnable artifacts

- **Artifact 1** (reproduced): Standard-library Python reconstructs the published matrix, checks every row pair, and reproduces the paper's autocorrelation exceptions and Gram distribution. [2](#reference-2)

### Computational notes

- A normalized candidate has 444889 unfixed signs after its first row and column are set to one. An exact validator must check 222778 unordered row-pair dot products, each of which must equal zero.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `hadamard-order-668`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Epoch AI, Hadamard Matrices, FrontierMath open-problem record (checked 24 July 2026). The source identifies order 668 as the smallest order divisible by four for which no real Hadamard matrix is known. https://epoch.ai/frontiermath/open-problems/hadamard
   - Also cited at Hadamard Matrices, accessed 2026-07-24
   - scholarly_publication; reference source; web version checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - Records 668 as the smallest multiple of four for which no real Hadamard matrix is known.
   - For A Hadamard matrix of order 668: A concise, independently checkable formulation of the smallest unresolved Hadamard order.
2. <a id="reference-2"></a>Shalom Eliahou, A 64-modular Hadamard matrix of order 668, Australasian Journal of Combinatorics 93(2) (2025), 422-427. Theorem 2.3, Fact 3.1, and the Gram-matrix statistics on pp. 424-426 https://ajc.maths.uq.edu.au/pdf/93/ajc_v93_p422.pdf
   - Also cited at Shalom Eliahou, A 64-modular Hadamard matrix of order 668, Australasian Journal of Combinatorics 93(2) (2025), 422-427, abstract, Introduction, and Conclusion
   - Also cited at Eliahou 2025, Fact 3.1 and the Gram-matrix statistics on page 426
   - journal_article; primary source; version of record; checked 2026-07-24
   - Source use: citation_only
   - Constructs a 64-modular order-668 matrix and gives the Gram data showing that it is not a real Hadamard matrix.
3. <a id="reference-3"></a>Matteo Cati and Dmitrii V. Pasechnik, A database of constructions of Hadamard matrices, arXiv:2411.18897v2 (2024). Matteo Cati and Dmitrii V. Pasechnik, A database of constructions of Hadamard matrices, arXiv:2411.18897v2: page 1, lines listing unknown orders; Section 3, page 5, the order-668 SageMath query; Appendix Tables 2 and 4, pages 15 and 17 https://arxiv.org/abs/2411.18897
   - Also cited at Cati and Pasechnik, revised 2025-08-30, pages 1, 5, 15, and 17
   - Also cited at Cati and Pasechnik, arXiv:2411.18897v2, Sections 1, 3, and 6 plus Appendix Tables 2 and 4; Eliahou 2025, Fact 3.1 and Conclusion; Kharaghani and Tayfeh-Rezaie 2005, DOI 10.1002/jcd.20043
   - preprint; reference source; arXiv:2411.18897v2; checked 2026-07-24
   - Source use: citation_only
   - Catalogues current Hadamard constructions and leaves order 668 without a real construction.
   - Source named by the research packet.
4. <a id="reference-4"></a>Andriyan Bayu Suksmono, A quantum approximate optimization method for finding Hadamard matrices, Scientific Reports 15 (2025), Article 33254. Introduction and the order-668 discussion https://www.nature.com/articles/s41598-025-18778-1
   - journal_article; primary source; version of record; checked 2026-07-24
   - Source use: citation_only
   - Discusses a quantum approximate search at order 668 while retaining the exact existence question as open.