  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.
