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The conventional maximum a posteriori receiver for coded code-division multiple-access (CDMA) systems has exponential computational complexity in terms of the number of users and the memory of the channel code. In this letter, we propose a low-complexity soft-input soft-output (SISO) multiuser detector based on the reduced-state a posteriori probability algorithm. Per-survivor processing and soft interference cancellation are used to remove the residual past and future interference in the branch metric computation. The complexity of the proposed receiver is related to the reduced memory of the CDMA channel and can be adjusted according to the complexity/performance tradeoff. Simulation results show that for asynchronous convolutionally coded systems, the proposed receiver can achieve the near-single-user performance for moderate to high signal-to-noise ratios.