  Associated to any finite metric space are a large number of objects and
quantities which provide some degree of structural or geometric information
about the space. In this paper we show that in the setting of subsets of
weighted Hamming cubes there are unexpected relationships between many of these
quantities. We obtain in particular formulas for the determinant of the
distance matrix, the $M$-constant and the cofactor sum for such spaces. This
work extends previous results for unweighted metric trees, and more generally,
for subsets of standard Hamming cubes, as well as results for weighted metric
trees. Finally we consider polygonal equalities in these spaces, giving a
complete description of nontrivial $1$-polygonal equalities that can arise in
weighted Hamming cubes.
