Finding many D-optimal designs by randomised decomposition and switching

The Hadamard maximal determinant (maxdet) problem is to find the maximum determinant H(n) of a square {+1,−1}-matrix of given order n. Such a matrix with maximum determinant is called a Doptimal design of order n. We consider some cases where n 6= 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 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 D-optimal designs (for order 26) and new conjectured D-optimal designs (for orders 27 and 33).