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This paper proposes a capacity-approaching, yet simple scheme for multi-input multiple-output (MIMO) channels. The proposed scheme is based on a concatenation of a mixture of short memory-length convolutional codes or repetition codes and a short, and simple rate-1 linear block code, followed by either 1-dimensional (1-D) anti-Gray or Gray mapping of quadrature phase-shift keying (QPSK) modulation. By interpreting the rate-1 code and the 1-D mapping as a multi-D mapping performed over multiple transmit antennas, the error performance is analyzed in two regions. In the error-floor region, a tight union bound and the corresponding design criterion on the asymptotic performance are derived. The bound provides a useful tool to predict the error performance at relatively low bit error rate (BER) values. Based on the obtained design criterion, an optimal rate-1 code for each 1-D mapping is then constructed to achieve the best asymptotic performance. In the turbo pinch-off region, by using extrinsic information transfer (EXIT) charts, the most suitable mixed codes are selected for both symmetric and asymmetric antenna configurations. It is demonstrated that the simple concatenation scheme can achieve a near-capacity performance over the MIMO channels. Furthermore, its error performance is shown to be comparable to that obtained by using well-designed irregular LDPC and RA codes, and therefore, the proposed scheme significantly outperforms a scheme employing a parallel concatenated turbo code. Simulation results in various cases are provided to verify the analysis.