# Order 37

## Some saturated D-optimal designs of order 37

There are many Hadamard equivalence classes of saturated
D-optimal designs (that is to say, maximal determinant {+1, -1} matrices) of
order 37.
Orrick and Solomon (2 May 2003) found the first, which we call
R,
and it was proved to be maximal by
Brent, Osborn, Orrick and Zimmermann (7 Aug 2009).
The maximal determinant for order 37 is
72*9^17*2^36 = 2^39*3^36
(93.63% of the Ehlich bound).

As discussed in our paper, there is a unique (up to symmetric signed
permutations) Gram matrix G = R.R^T = R^T.R.

For G and R, see
Will Orrick's page.

Other saturated D-optimal designs (maxdet matrices) of order 37 can be found
by randomised decomposition of the Gram matrix G. So far we have found 100
HT-equivalence classes of such matrices (including R), but there are
certainly many more than this. A list of known HT-equivalence classes
of solutions is here.
By taking duals (transposes) you can obtain twice as many Hadamard
equivalence classes.

## Candidate Gram matrices

As discussed in our paper, the proof that R has maximal determinant
depends on testing 807 candidate Gram matrices. These are given
here in compressed format
(one matrix per line). Further information about these 807
candidate Gram matrices may be found here.
They fall into 489 equivalence classes (where equivalence of two matrices
means that they have the same characteristic polynomial), with 284
distinct determinants. We checked 1528 pairs of equivalent matrices.