Minimum probability of error of many-user CDMA without power control

The minimum-probability-of-error (optimal) detector of multiuser CDMA is shown to be equivalent to a conditional mean estimator which finds the mean value of the stochastic output of a so-called Bayes retrochannel. The equivalence of the Bayes retrochannel to a spin glass allows the optimal bit-error-rate to be determined exactly in the large-system limit by studying macroscopic properties of the spin glass using tools developed in statistical mechanics. It is found that all users, regardless of their individual received energies, enjoy the same multiuser efficiency, which is directly related to the mean-square error of the conditional mean estimator.