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The jointly optimum noncoherent multiuser detector is obtained for nonlinear nonorthogonal modulation over the frequency nonselective Rayleigh-fading multiple-access channel. Upper and lower bounds on the average bit-error probability are derived. While these bounds are numerically computable, they are too complicated to give insight into the relative influence of the system parameters on the essential behavior of the bit-error rate. Hence this paper develops an asymptotic analysis of the average bit-error probability. In particular, it is shown that the upper and lower bounds are asymptotically convergent. An exact formula for the asymptotic efficiency of the optimum noncoherent detector is derived. Interestingly, the asymptotic efficiency is found to be positive and independent of the signal strengths of the interfering users. In contrast, the noncoherent detector which would be optimal in a single-user channel (the "conventional detector"), when used over the multiuser channel, has an asymptotic efficiency that is identically equal to zero no matter what the powers of the interferers may be. While the performance analysis of the optimum detector provides the fundamental limit on achievable error rate, the implementational complexity of the optimum detector is exponential in the number of users. As a low-complexity alternative, a decorrelative energy detector is also proposed and analyzed in terms of error probability and asymptotic efficiency.