Binary linear codes via 4D discrete Ihara-Selberg function

Abstract

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 the 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 the cubic 3D lattices.

Keywords

• Graph polynomial
• hyper-matrix
• hyper-determinant
• Bass' theorem
• Ising partition function
• binary linear code
• weight enumerator
• discrete Ihara-Selberg function