A distributed source coding problem with a joint distortion criterion that depends on the sources and the reconstruction is considered in this work. 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, a new achievable rate-distortion region (an inner bound to the performance limit) is presented for this problem for discrete memoryless sources based on “good” structured random nested codes built over abelian groups. For certain sources and distortion functions, the new rate region is shown to be strictly bigger than the Berger-Tung rate region, which has been the best known achievable rate region for this problem till now. This is done using numerical plots. Achievable rates for single-user source coding using abelian group codes are also obtained as a corollary of the main coding theorem. It is shown that nested linear codes achieve the Shannon rate-distortion function in the arbitrary discrete memoryless case.