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

Ralf Koetter, Wen Ching W. Li, Pascal O. Vontobel, Judy L. Walker

Research output: Contribution to journalArticle

36 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)205-229
Number of pages25
JournalAdvances in Mathematics
Volume213
Issue number1
DOIs
StatePublished - Aug 1 2007

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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