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A result of Robinson states that no $OD(n; 1, 1, 1, 1, 1, n-5)$
exists for $n>40$. We complement this result by showing the
existence of $OD(n; 1, 1, 1, 1, 1, n-5)$ for $n=32, 40$. This
includes a resolution to an old open problem regarding orthogonal
designs of order 32 as well. We also obtain a number of new
orthogonal designs of order 32.
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