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The 2times2 MIMO profiles included in mobile WiMAX specifications are Alamouti's space-time code (STC) for transmit diversity and spatial multiplexing (SM). The former has full diversity and the latter has full rate, but neither of them has both of these desired features. An alternative 2times2 STC, which is both full rate and full diversity, is the Golden code. It is the best known 2times2 STC, but it has a high decoding complexity. Recently, the attention was turned to the decoder complexity, this issue was included in the STC design criteria, and different STCs were proposed. In this paper, we first present a full-rate full-diversity 2times2 STC design leading to substantially lower complexity of the optimum detector compared to the Golden code with only a slight performance loss. We provide the general optimized form of this STC and show that this scheme achieves the diversity multiplexing frontier for square QAM signal constellations. Then, we present a variant of the proposed STC, which provides a further decrease in the detection complexity with a rate reduction of 25% and show that this provides an interesting trade-off between the Alamouti scheme and SM.