Weighting matrices appear in the statistical design of experiments (Weighting Matrices and Statistical Design of Experiments.

A weighting matrix is a generalization of Hadamard matrices, e.g.,

HadamardMatrix[8] HadamardMatrix[8].Inverse[HadamardMatrix[8]] HadamardMatrix[5] 

but can have zeroes as entries in addition to -1 and +1--and they exist for square matrices that are not 4*(positive integer).

Beyond a random search which is trivial and not a great method, does anyone know how to compute a Weighing matrix of size (general positive integer)?