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Annales de l’Institut Henri Poincaré D

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Volume 6, Issue 1, 2019, pp. 73–95
DOI: 10.4171/AIHPD/65

Published online: 2018-07-25

Binary linear codes via 4D discrete Ihara–Selberg function

Martin Loebl[1]

(1) Charles University, Prague, Czechia

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 1960s 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

Loebl Martin: Binary linear codes via 4D discrete Ihara–Selberg function. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 6 (2019), 73-95. doi: 10.4171/AIHPD/65