15-00-HadamardConjecture.tex

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\pmtitle{Hadamard conjecture}
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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 \cite{KT}.

\PMlinkexternal{See here}{http://math.ipm.ac.ir/tayfeh-r/papersandpreprints/h428.pdf}.

A Hadamard matrix of order 764 has also recently been constructed \cite{DZD}.

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.

\begin{thebibliography}{99}
\bibitem{KT} H. Kharaghani, B. Tayfeh-Rezaie, \emph{A Hadamard matrix of order 428}, J. Comb. 
Designs \textbf{13}, (2005), 435-440.
\bibitem{DZD} D.Z. Dokovi\'c, \emph{Hadamard matrices of order 764 exist}, \PMlinkexternal{preprint}{http://arxiv.org/abs/math/0703312v1}.
\end{thebibliography}
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