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Complex orthogonal space-time block codes (STBCs) with linear processing for more than two transmit antennas cannot achieve a full rate in multiple-input and multiple-output (MIMO) channels. However, at the expense of losing some degree of diversity advantage, it is possible to achieve a full rate or even rates higher than one for any number of transmit antennas. In this letter, we propose new closed-loop extended orthogonal STBCs (EO-STBCs) for three and four transmit antennas. We show that the schemes can achieve a full transmit diversity with linear processing, and they outperform the previous closed-loop STBCs. In addition, we investigate the closed-loop EO-STBCs with receive antenna selection. It is shown that for four transmit antennas and up to three receive antennas, performance gains of up to 5.8 dB are obtained over the closed-loop EO-STBCs without antenna selection.