  We consider a noisy Slepian-Wolf problem where two correlated sources are
separately encoded (using codes of fixed rate) and transmitted over two
independent binary memoryless symmetric channels. The capacity of each channel
is characterized by a single parameter which is not known at the transmitter.
The goal is to design systems that retain near-optimal performance without
channel knowledge at the transmitter.
  It was conjectured that it may be hard to design codes that perform well for
symmetric channel conditions. In this work, we present a provable
capacity-achieving sequence of LDGM ensembles for the erasure Slepian-Wolf
problem with symmetric channel conditions. We also introduce a staggered
structure which enables codes optimized for single user channels to perform
well for symmetric channel conditions.
  We provide a generic framework for analyzing the performance of joint
iterative decoding, using density evolution. Using differential evolution, we
design punctured systematic LDPC codes to maximize the region of achievable
channel conditions. The resulting codes are then staggered to further increase
the region of achievable parameters. The main contribution of this paper is to
demonstrate that properly designed irregular LDPC codes can perform well
simultaneously over a wide range of channel parameters.
