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This study deals with a multiple-access transmission scheme based on random permutations, which allows one to obtain both spectrum spreading and time spreading. In particular, one is interested by the multi-user detection when the transmission is achieved in the context of asynchronous frequency-selective and time-varying channels. For this detection, it is assumed that the channel coefficients and delays are unknown at the receiver side. Instead, only rough estimates of the minimum and maximum delays are available. However, it is supposed that the transmitter sends pilot signals, that is, training bit sequences, which allow the receiver to perform an LMS (least-mean-square)-type detection. This study provides theoretical performance results. Moreover, this random permutation-based transmission scheme is compared with the DS-CDMA system, for which a similar LMS receiver is defined, under the same assumptions. It is then shown that, for time-varying channels, the permutation-based system presents lower bit-error rates than the DS-CDMA system, since the former provides time diversity.