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In this paper, we consider multicarrier systems with multiple transmit antennas under a power-balancing constraint. Applying transmit antenna selection and discrete rate-adaptive modulation using M-ary quadrature-amplitude modulation (QAM) according to the channel variation per subcarrier, we develop an optimal transmit antenna selection scheme in terms of the maximum spectral efficiency, where all the possible groupings for sending the same information-bearing signals in a group of subcarriers are searched, and the groups of subcarriers for providing the frequency diversity gain are formed. In addition, we propose a suboptimal method for reducing the computational complexity of the optimal method. The suboptimal scheme considers only the subcarriers under outage, and these subcarriers are sequentially combined until the required signal-to-noise ratio (SNR) is met. Numerical results show that the proposed suboptimal method with diversity combining outperforms the optimal antenna selection without diversity combining, as introduced in the work of Sandell and Coon, particularly for low-SNR regions, and offers the spectral efficiency close to the optimal method with diversity combining while maintaining lower complexity.