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].

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 $H_n$ 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.

Bibliography

1

H. Kharaghani, B. Tayfeh-Rezaie, A Hadamard matrix of order 428, J. Comb. Designs 13, (2005), 435-440.

2

D.Z. Dokovic, Hadamard matrices of order 764 exist, preprint.