  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.
