Binary Linear Codes, Dimers and Hypermatrices
arXiv:1302.1722
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.
To appear in the proceedings of 10th Random Generation of Combinatorial Structures (GASCom) (2016). 10 pages, 6 figures