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Recently, Guo and Xia introduced low complexity decoders called Partial Interference Cancellation (PIC) and PIC with Successive Interference Cancellation (PIC-SIC), which include the Zero Forcing (ZF) and ZF-SIC receivers as special cases, for point-to-point MIMO channels. In this paper, we show that PIC and PIC-SIC decoders are capable of achieving the full cooperative diversity available in wireless relay networks. We give sufficient conditions for a Distributed Space-Time Block Code (DSTBC) to achieve full diversity with PIC and PIC-SIC decoders and construct a new class of DSTBCs with low complexity full-diversity PIC-SIC decoding. We also show that almost all known full-diversity PIC/PIC-SIC decodable codes constructed for point-to-point MIMO channels can be used as full-diversity PIC/PIC-SIC decodable DSTBCs in relay networks. The proposed DSTBCs with low complexity, full-diversity PIC/PIC-SIC decoding achieve higher rates (in complex symbols per channel use) ...