Hadamard conjecture

There exists a Hadamard matrixMathworldPlanetmath of order n = 4m, for all m+.

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 Hn when n is divisible by 4 and of the form 2e(pm+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:


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


  • 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