In this paper, we consider the distributed source coding problem with a joint distortion criterion. We use random structured codes, specifically nested Abelian group codes to achieve a new inner bound to the rate-distortion region. This new inner bound unifies all other known results on the distributed source coding problem and for certain sources, this inner bound is strictly larger than other known rate regions. Furthermore, we define the notion of “nested random/group codes” in which the inner code is a group code and the outer code is a random code such that the quantization operation is a random vector quantization and the binning operation is correlated. We use random/group codes to improve upon the rate region achieved using nested group codes. We define two fundamental quantities of Abelian group codes: the “channel coding group entropy” and the “source coding group entropy” to present the achievable rate region.