There exists a Hadamard matrix of order $n$ = $4m$, for all $m\in\mathbb{Z}^{+}.$

A Hadamard matrix of order 428 (m=107) has been recently constructed [1].

http://math.ipm.ac.ir/tayfeh-r/papersandpreprints/h428.pdfSee here.

A Hadamard matrix of order 764 has also recently been constructed [2].

Also, Paley’s theorem guarantees that there always exists a Hadamard matrix $H_{n}$ when $n$ is divisible by 4 and of the form $2^{e}(p^{m}+1)$, for some positive integers e and m, and p an odd prime and the matrices can be found using Paley construction.

This leaves the order of the lowest unknown Hadamard matrix as 668. There are 13 integers $m$ less than 500 for which no Hadamard matrix of order $4m$ is known:

 $167,179,223,251,283,311,347,359,419,443,479,487,491$

and all of them are primes congruent to 3 mod 4.

## References

• 1 H. Kharaghani, B. Tayfeh-Rezaie, A Hadamard matrix of order 428, J. Comb. Designs 13, (2005), 435-440.
• 2 D.Z. Doković, Hadamard matrices of order 764 exist, http://arxiv.org/abs/math/0703312v1preprint.
Title Hadamard conjecture HadamardConjecture 2013-03-22 14:07:07 2013-03-22 14:07:07 Mathprof (13753) Mathprof (13753) 12 Mathprof (13753) Conjecture msc 15-00 Hadamard’s conjecture HadamardMatrix