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We introduce a new family of space-time codes called embedded Alamouti space-time (EAST) codes which offer high rate, a nonvanishing determinant and low decoding complexity. The family is parameterized by both the number of transmit antennas, which can range from two to eight, and by the rate, which can range from one to half the number of transmit antennas. The EAST codes combine a modified version of the perfect space-time codes with an Alamouti embedding. For rates higher than one, the resulting space-time codes are simultaneously lower in decoding complexity and better performing than all known previous constructions in terms of the error probability achieved on a quasistatic Rayleigh fading channel with a given dimension.