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In this paper, we investigate the benefit of joint network-channel (JNC) coding for the multiple-access full-duplex relay channel. We use superposition block Markov network-channel (distributed) coding and (iterative) joint decoding at the destination. Prominent amongst our system assumptions are the fact that the medium access is not orthogonal and that the source-to-relay links are not error-free, which render the reality of wireless environments and notably complicate the problem. Indeed, due to the broadcast nature of wireless media, transmitted signals interfere at the relay and at the destination and form multiple-access channels. This entails the need of sophisticated joint multiuser detection and decoding procedures. Moreover, improper (erroneous) distributed codes may result from unsuccessful decoding at the relay. This second issue is handled with selection relaying. We describe in detail the structure of the encoders, when and how JNC coding is performed, and the structure of the joint multiuser detectors and decoders. Finally, we evaluate the performance of the proposed schemes and compare them with conventional joint network-channel distributed coding for the orthogonal multiple-access half-duplex relay channel.