Binary linear codes, dimers and hypermatrices
2017
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. Keywords: Ising problem, dimer problem, partition function, perfect matching, Kasteleyn matrix, linear binary code, permanent, 3-dimensional cubic lattice, triangular configuration, 2-dimensional simplicial complex.
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