We study a multiuser detection system for code-division multiple access (CDMA). We show that applying multistage hard-decision parallel interference cancellation (HD-PIC) significantly improves performance compared to the matched filter system. In (multistage) HD-PIC, estimates of the interfering signals are used iteratively to improve knowledge of the desired signal. We use large deviation theory to show that the bit-error probability (BEP) is exponentially small when the number of users is fixed and the processing gain increases. We investigate the exponential rate of the BEP after several stages of HD-PIC. We propose to use the exponential rate of the BEP as a measure of performance, rather than the signal-to-noise ratio (SNR), which is often not reliable in multiuser detection models when the system is lightly loaded. We show that the exponential rate of the BEP remains fixed after a finite number of stages, resulting in an optimal hard-decision system. When the number of users becomes large, the exponential rate of the BEP converges to (log 2)/2 $1/4. We provide guidelines for the number of stages necessary to obtain this asymptotic exponential rate. We also give Chernoff bounds on the BEPs. These estimates show that the BEPs are quite small as long as k = o(n/log n) when the number of stages of HD-PIC is fixed, and even exponentially small when k = O(n) for the optimal HD-PIC system, and where k is the number of users in the system and n is the processing gain. Finally, we extend the results to the case where the number of stages depends on k in a certain manner. The above results are proved for a noiseless channel, and we argue that we expect similar results in a noisy channel as long as the two-sided spectrum of the noise decreases proportionally to n.