Binary Linear Codes, Dimers and Hypermatrices

TL;DR

This paper establishes a novel connection between binary linear codes, hypermatrices, and dimer models by expressing weight enumerators and partition functions as determinants and permanents of 3-dimensional hypermatrices.

Contribution

It introduces a new representation of binary linear code weight enumerators as hypermatrix permanents and relates dimer partition functions to determinants of 3-matrices.

Findings

Weight enumerator equals the permanent of a 3-matrix.

Each permanent can be expressed as a determinant of a 3-matrix.

Dimer partition function can be written as a determinant of a vertex-adjacency 3-matrix.

Abstract

We show that the weight enumerator of any binary linear code is equal to the permanent of a 3-dimensional hypermatrix (3-matrix). We also show that each permanent is a determinant of a 3-matrix. As an application we write the dimer partition function of a finite 3-dimensional cubic lattice as the determinant of the vertex-adjacency 3-matrix of a 2-dimensional simplicial complex which preserves the natural embedding of the cubic lattice.