How to Find a Weighing Matrix of Size n?

Question

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)?

Answer 1

Here is a straightforward approach for small (less than $4$) n and w:

ClearAll[a, A]; A = Table[a[i, j], {i, 4}, {j, 4}];
FindInstance[A . Transpose[A] == 2*IdentityMatrix[4] && Flatten[A] >= -1 && 
Flatten[A] <= 1, Flatten[A], Integers]

{{a[1, 1] -> -1, a[1, 2] -> -1, a[1, 3] -> 0, a[1, 4] -> 0, a[2, 1] -> -1, a[2, 2] -> 1, a[2, 3] -> 0, a[2, 4] -> 0, a[3, 1] -> 0, a[3, 2] -> 0, a[3, 3] -> -1, a[3, 4] -> 1, a[4, 1] -> 0, a[4, 2] -> 0, a[4, 3] -> -1, a[4, 4] -> -1}}

For many n and k a weighting matrix does not exist and the above code derives it. For example, FindInstance[ A . Transpose[A] == 5*IdentityMatrix[4] && Flatten[A] >= -1 && Flatten[A] <= 1, Flatten[A], Integers] results in {}.