This correspondence highlights the performance limitation of linear minimum mean-squared error (mmse) detection in underdetermined vector Gaussian channels (as in overloaded code-division multiple-access (CDMA) systems) where the number of symbols (users) exceeds the signal space dimension (spread factor). It is shown that for such a simple receiver it is not possible to construct signal sets (or spreading codes) to even satisfy the basic requirement that every user's symbol error probability

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... cays exponentially as noise power vanishes. This result holds for arbitrary received energies, modulation schemes, and any strictly underdetermined system with a finite signal space dimension and a finite number of users. Index Terms-Code-division multiple-access (CDMA), Gaussian multiple-access channels, multiuser detection, minimum mean square error (mmse) detection, signal design.