Existence of some signed magic arrays

We consider the notion of a signed magic array, which is an m×n rectangular array with the same number of filled cells s in each row and the same number of filled cells t in each column, filled with a certain set of numbers that is symmetric about the number zero, such that every row and column has a zero sum. We attempt to make progress toward a characterization of for which (m,n,s,t) there exists such an array. This characterization is complete in the case where n=s and in the case where n=m; we also characterize three-fourths of the cases where n=2m.