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.Comment: The third version contains several small updates suggested by
  reviewer