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This paper considers an amplify and forward DS/CDMA cooperative relay system with half-duplex relays. Given different receive structures for this scenario, the asymptotic spectral efficiency is analyzed assuming that the spreading factor and the information block length go to infinity at the same rate. The objective of this analysis is to identify the cases where transmission diversity, introduced by the relays, compensates for the physical-rate loss caused by the fact that the source terminal remains silent during the relay transmission. Results show that, provided that the source-destination channel amplitude is sufficiently low, there exists an optimum assignment of transmission time between source and relays.