Binary linear codes via 4D discrete Ihara-Selberg function

We express the weight enumerator of each binary linear code, in particular the Ising partition function of an arbitrary finite graph, as a formal infinite product. An analogous result was obtained by Feynman and Sherman in the beginning of the 1960's for the special case of the Ising partition function of planar graphs. A product expression is an important step towards understanding the logarithm of the Ising partition function, for general graphs and in particular for cubic 3D lattices.