Binary linear codes via 4D discrete Ihara-Selberg function

TL;DR

This paper presents a novel infinite product expression for the weight enumerator of binary linear codes and the Ising partition function of arbitrary graphs, advancing understanding of these functions beyond planar cases.

Contribution

It extends the product expression for the Ising partition function from planar graphs to general graphs, including 3D lattices, providing new insights into their structure.

Findings

Derived a formal infinite product for the weight enumerator of binary linear codes.

Extended Feynman and Sherman's results from planar to arbitrary graphs.

Facilitates analysis of the Ising partition function's logarithm for complex graphs.

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 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.