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The performance of iterative multiuser receivers in a turbo-coded asynchronous direct-sequence code-division multiple-access system over a fully interleaved Rayleigh flat-fading channel is presented. The full-complexity soft-in/soft-out (SISO) multiuser detector based on the logarithmic maximum a posteriori probability (log-MAP) algorithm has a computational complexity growing exponentially with the number of users. It is shown that the computational complexity can be significantly reduced by replacing the MAP multiuser detector in the receiver with suboptimal SISO detectors. A novel hybrid reduced-state algorithm that effectively combines the advantages of the T-algorithm and the channel memory reduction algorithm is proposed. The computational complexity of the proposed detector is determined by the reduced channel memory length and can be further reduced adaptively based on channel conditions. Simulation results show that a substantial reduction in complexity can be obtained at the expense of slight degradation in performance compared with the full-complexity MAP detector. The proposed algorithm is compared with other reduced-complexity schemes and the superiority of the proposed algorithm in terms of performance/complexity tradeoffs is demonstrated.