Starting with a set of square matrices with elements {-1, +1}, let their size be an even number, but not divisible by four. From this set of matrices, find the one with the maximal determinant, and you have a D-optimal matrix. Although simple for small matrices, the process becomes challenging once an exhaustive search is no longer computationally feasible.

The biggest reduction in search space comes from using two circulant matrices, each half the width of the D-optimal matrix you wish to find. A circulant matrix is square and consists of a row of elements that is right-shifted for each subsequent row. The elements that are pushed off the end are recycled to the beginning of the row. This reduces the problem down from quadratic in complexity to linear in the number of rows. From this point on, the solution is typically found by solving a system of linear and quadratic equations based on conditions that must hold for the two circulant matrices.

The authors of this paper recast this final step as a linear and quadratic integer optimization problem. Symmetry in the periodic autocorrelation functions and constraints on the power spectral density are taken from another paper and combined with two further types of symmetry. The authors then show that if the size of the circulant matrices is taken to be a multiple of three, even further constraints can be placed on the elements of the two circulant matrices. Finally, the authors show that two known D-optimal matrices do satisfy these extra constraints. Why would you want to find a D-optimal matrix? Well, the authors do not make that point entirely clear.