Hadamard conjecture
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 |
---|---|
Canonical name | HadamardConjecture |
Date of creation | 2013-03-22 14:07:07 |
Last modified on | 2013-03-22 14:07:07 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 12 |
Author | Mathprof (13753) |
Entry type | Conjecture |
Classification | msc 15-00 |
Synonym | Hadamard’s conjecture |
Related topic | HadamardMatrix |