Graph-Cover Decoding and Finite-Length Analysis of Message-Passing Iterative Decoding of LDPC Codes

TL;DR

This paper introduces graph-cover decoding as a new framework for analyzing finite-length message-passing decoding of LDPC codes, linking it to linear programming decoding and the fundamental polytope.

Contribution

It presents the concept of graph-cover decoding, establishing its connection to linear programming decoding and providing insights into the fundamental polytope for finite-length analysis.

Findings

Graph-cover decoding is equivalent to linear programming decoding.

The fundamental polytope characterizes message-passing decoding behavior.

Connections between graph covers and decoding performance are elucidated.

Abstract

The goal of the present paper is the derivation of a framework for the finite-length analysis of message-passing iterative decoding of low-density parity-check codes. To this end we introduce the concept of graph-cover decoding. Whereas in maximum-likelihood decoding all codewords in a code are competing to be the best explanation of the received vector, under graph-cover decoding all codewords in all finite covers of a Tanner graph representation of the code are competing to be the best explanation. We are interested in graph-cover decoding because it is a theoretical tool that can be used to show connections between linear programming decoding and message-passing iterative decoding. Namely, on the one hand it turns out that graph-cover decoding is essentially equivalent to linear programming decoding. On the other hand, because iterative, locally operating decoding algorithms like message-passing iterative decoding cannot distinguish the underlying Tanner graph from any covering graph, graph-cover decoding can serve as a model to explain the behavior of message-passing iterative decoding. Understanding the behavior of graph-cover decoding is tantamount to understanding the so-called fundamental polytope. Therefore, we give some characterizations of this polytope and explain its relation to earlier concepts that were introduced to understand the behavior of message-passing iterative decoding for finite-length codes.