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Blind multiuser detection is studied in the presence of multipath distortion. The kurtosis of the receiver's output is minimized with respect to the receiver's parameters subject to multiple linear constraints. The constraints are dependent on the spreading codes of the user of interest. In order to combine multipath signals optimally, the kurtosis is parameterized by the constraint vector as well and is further maximized. It is shown that under some conditions, the optimal constraint vector converges to the channel vector of the desired user irrespective of noise, and the proposed receiver ensures perfect cancellation of both intersymbol interference and multiuser interference. Meanwhile, a minimum mean-square-error receiver can be constructed from the constraint vector.