On Optimal Weighing Designs

There are two criteria for optimality of weighing designs. One, which has been widely studied, is that the determinant of \(XX^T\) should be maximal, where \({X}\) is the weighing matrix. The other is that the trace of \((XX^T)^{-1}\) should be minimal. We examine the second criterion. It is shown that Hadamard matrices, when they exist, are optimal with regard to the second criterion, just as they are for the first one.