On harmonic weight enumerators of binary codes

We define some new polynomials associated to a linear binary code and a har monic function of degree k. The case k=0 is the usual weight enumerator of the code. When divided by (xy)(k), they satisfy a MacWilliams type equality . When applied to certain harmonic functions constructed from Hahn polynomi als, they can compute some information on the intersection numbers of the c ode. As an application, we classify the extremal even formally self-dual co des of length 12.