Point configurations in finite or compact infinite metric spaces satisfy positivity constraints arising from the properties of positive definite functions on the space. We introduce the general framework of the method in the case of homogeneous spaces and consider in detail the example of the Hamming space. Other examples include the sphere in R^n, the "Johnson space," etc. We discuss the computation of bounds on the size of a maximum packing (code) and derive new bounds on the maximum size of a code with a given number of distinct distances.