In this work, we consider a distributed source coding problem with a
joint distortion criterion depending on the sources and the
reconstruction. This includes as a special case the problem of
computing a function of the sources to within some distortion and also
the classic Slepian-Wolf problem, Berger-Tung
problem, Wyner-Ziv problem,
Yeung-Berger problem and the Ahlswede-Korner-Wyner
problem. While the prevalent trend in
information theory has been to prove achievability results using
Shannon's random coding arguments, using structured random codes offer
rate gains over unstructured random codes for many problems. Motivated
by this, we present a new achievable rate-distortion region 
for this problem for discrete
memoryless sources based on ``good'' structured random nested codes
built over abelian groups. We demonstrate rate gains for this problem
over traditional coding schemes using random unstructured codes. For
certain sources and distortion functions, the new rate region is
strictly bigger than the Berger-Tung rate region, which has been the
best known achievable rate region for this problem till  now.
Further, there is no known unstructured random coding scheme that
achieves these rate gains. Achievable performance limits for
single-user source coding using abelian group codes are also obtained
as parts of the proof of the main coding theorem. As a corollary, we
also prove that nested linear codes achieve the Shannon
rate-distortion bound in the single-user setting. Note that while
group codes retain some structure, they are more general than linear
codes which can only be built over finite fields which are known to
exist only for certain sizes.