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In this paper, a new method is proposed to obtain full-diversity, rate-2 (rate of two complex symbols per channel use) space-time block codes (STBCs) that are full-rate for multiple input double output (MIDO) systems. Using this method, rate-2 STBCs for 4 × 2, 6 × 2, 8 × 2, and 12 × 2 systems are constructed and these STBCs are fast ML-decodable, have large coding gains, and STBC-schemes consisting of these STBCs have a non-vanishing determinant (NVD) so that they are DMT-optimal for their respective MIDO systems. It is also shown that the Srinath-Rajan code for the 4 × 2 system, which has the lowest ML-decoding complexity among known rate-2 STBCs for the 4 × 2 MIDO system with a large coding gain for 4-/16-QAM, has the same algebraic structure as the STBC constructed in this paper for the 4 × 2 system. This also settles in positive a previous conjecture that the STBC-scheme that is based on the Srinath-Rajan code has the NVD property and hence is DMT-optimal for the 4 × 2 system.