The detection of a weak stochastic signal in multi-variate impulsive noise is investigated. The signal is that of a new user in a Code-Division Multiple-Access multi-user system. A particular type of transformation noise is considered: the trasmitted signal vector is modified by a linear transformation prior to signal detection thereby yielding a signal process and noise process that are dependent from component to component of the recieved vector. The linear operator is designed to combat multiple-access interference. The effect of such a transformation is studied by deriving the efficacies and asymptotic relative efficiencies of several detectors. Both linear and non-linear cross-correlator detector structures are examined. The performance of the detectors is determined for two types of non-Gaussian noise: additive channel noise and residual multiple-access interference noise. The numerical results are consistent with previous findings and motivate the investigation of a hybrid detector which is composed of a combination of the linear correlator and the polarity coincidence correlator. The asymptotic normality of the test statistics under study is also investigated.