Finding D-optimal designs by randomised decomposition and switching

A square {+1,-1}-matrix of order n with maximal determinant is called a saturated D-optimal design. We consider some cases of saturated Doptimal designs where n > 2, n ≢ 0 mod 4, so the Hadamard bound is not attainable, but bounds due to Barba or Ehlich and Wojtas may be attainable. If R is a matrix with maximal (or conjectured maximal) determinant, then G = RR^T is the corresponding Gram matrix. For the cases that we consider, maximal or conjectured maximal Gram matrices are known. We show how to generate many Hadamard equivalence classes of solutions from a given Gram matrix G, using a randomised decomposition algorithm and row/column switching. In particular, we consider orders 26, 27 and 33, and obtain new saturated D-optimal designs (for order 26) and new conjectured saturated D-optimal designs (for orders 27 and 33).