Characterizations of Pseudo-Codewords of LDPC Codes

TL;DR

This paper explores the fundamental cone associated with LDPC codes, providing properties and characterizations that relate to decoding algorithms and graph representations, especially for cycle codes.

Contribution

It introduces a characterization of the fundamental cone based on the parity check matrix and connects it to graph zeta functions for cycle codes.

Findings

Properties of the fundamental cone derived from graphical models

Connection between the fundamental polytope and the Newton polytope of the Hashimoto zeta function

Insights into decoding performance based on geometric representations

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.