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.