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A recent result of Schmidt has brought Williamson matrices back into
the spotlight. In this paper
a new algorithm is introduced to search for hard to
find Williamson matrices. We find all nonequivalent Williamson
matrices of odd order $n$ up to $n=59$. It turns out that there
are none for $n=35, 47, 53, 59$ and it seems that the Turyn class
may be the only infinite class of these matrices.
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