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Orthogonal space-time block-coded (OSTBC) orthogonal frequency-division multiplexing (OFDM) links for frequency-selective multiple-input multiple-output (MIMO) channels with correlated paths and transmit antennas are considered. In such systems, optimal precoding with only covariance feedback is derived using the minimum pair-wise error probability (PEP) criterion; linear and non-linear precoders are designed. The proposed precoding only needs the statistical knowledge of the channel at the transmitter, which significantly reduces the feedback requirements. Both linear and non-linear precoders substantially improve the system bit error rate (BER) for OSTBC OFDM in transmit-antenna and path-correlated channels. The proposed non-linear precoder outperforms the linear precoder.