Complex D-optimal designs (Q1923516)

Let \(g(n)\) (resp. \(\gamma(n)\)) be the maximum absolute value of the determinant of all \(n\times n\) matrices whose entries are from \(\{1,-1\}\) (resp. \(\{1,-1,i,-i\})\). The following theorems are proved: (1) \(g(2m)\geq 2^m(\gamma(m))^2\), with equality iff there exists a matrix of the form \((\begin{smallmatrix} A & B\\ -B & A\end{smallmatrix})\) for which \(g(2m)\) is attained (\(A\), \(B\) are \(m\times m\) matrices). (2) If \(m\) is odd, then \(\gamma(m)\leq (2m-1)^{1/2}(m-1)^{(m-1)/2}\), and inequality is impossible unless \(2m-1\) is the sum of two integer squares. A further necessary condition for equality as well as interesting conjectures are presented.