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A linear precoder for MIMO channels exploiting both the channel mean and transmit correlation has been shown to improve performance of an orthogonal space-time coded system (Vu, M. and Paulraj, A., Proc. IEEE Vehicular Tech. Conf., 2004). We extend the precoder design to systems with non-orthogonal space-time code, and provide asymptotic analysis at high K factor. The precoder is designed by minimizing the Chernoff bound on the pairwise error probability. While a linear precoder can be viewed as a multi-mode beamformer, it converges to a single beam as the K factor increases. A design criterion based on the minimum codeword distance and a new criterion based on the average codeword distance are considered. Numerical simulations using quasi-orthogonal STBC give examples of the performance gain that can be achieved with these designs.