  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.
