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Characterizations of pseudo-codewords of (low-density) parity-check codes

Authors
Journal
Advances in Mathematics
0001-8708
Publisher
Elsevier
Publication Date
Volume
213
Issue
1
Identifiers
DOI: 10.1016/j.aim.2006.12.010
Keywords
  • (Low-Density) Parity-Check Codes
  • Pseudo-Codewords
  • Fundamental Cone
  • Graph Covers
  • Zeta Functions
  • Newton Polyhedron
Disciplines
  • Computer Science

Abstract

Abstract An important property of low-density parity-check codes is the existence of highly efficient algorithms for their decoding. Many of the most efficient, recent graph-based algorithms, e.g. message-passing iterative decoding and linear programming decoding, crucially depend on the efficient representation of a code in a graphical model. In order to understand the performance of these algorithms, we argue for the characterization of codes in terms of a so-called fundamental cone in Euclidean space. This cone depends upon a given parity-check matrix of a code, rather than on the code itself. We give a number of properties of this fundamental cone derived from its connection to unramified covers of the graphical models on which the decoding algorithms operate. For the class of cycle codes, these developments naturally lead to a characterization of the fundamental cone as the Newton polyhedron of the Hashimoto edge zeta function of the underlying graph.

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