Characterizations of pseudo-codewords of LDPC codes

Authors

Ralf Koetter, University of Illinois at Urbana-ChampaignFollow
Wen-Cheng W. Li, Pennsylvania State University - Main CampusFollow
Pascal O. Vontobel, University of Wisconsin-MadisonFollow
Judy L. Walker, University of Nebraska - LincolnFollow

Document Type

Article

Date of this Version

2007

Citation

Published in Advances in Mathematics, 213 (2007) 205-229.

Archived version from http://arxiv.org/abs/cs/0508049v3

Updated: June 18, 2018

Comments

Copyright Elsevier. Used by permission.

Abstract

An important property of high-performance, low complexity codes is the existence of highly efficient algorithms for their decoding. Many of the most efficient, recent graph-based algorithms, e.g. message passing algorithms and decoding based on linear programming, 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 which is a function of a given parity check matrix of a code, rather than of 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 polytope as the Newton polytope of the Hashimoto edge zeta function of the underlying graph.