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There exists a Hadamard matrix of order  =  , for all
A Hadamard matrix of order 428 (m=107) has been recently constructed [1].
See 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  when  is divisible by 4 and of the form
 , 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  less than 500 for which no Hadamard matrix of order  is known:
and all of them are primes congruent 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, preprint.
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