Summary: We construct a new family of linear space-time (ST) block codes by the combination of rotated constellations and the Hadamard transform, and we prove them to achieve the full transmit diversity over a quasi-static or fast fading channel. The proposed codes transmit at a normalized rate of 1 symbol/\(s\). When the number of transmit antennas \(n=1\) or 2, or \(n\) is a multiple of four, we spread a rotated version of the information symbol vector by the Hadamard transform and send it over \(n\) transmit antennas and \(n\) time periods; for other values of \(n\), we construct the codes by sending the components of a rotated version of the information symbol vector over the diagonal of an \(n\times n\) ST code matrix. The codes maintain their rate, diversity, and coding gains for all real and complex constellations carved from the ring \(\mathbb Z[i]\), and they outperform the codes from orthogonal design when using complex constellations for \(n>2\). The maximum-likelihood decoding of the proposed codes can be implemented by the sphere decoder at a moderate complexity. It is shown that using the proposed codes in a multiantenna system yields good performances with high spectral efficiency and moderate decoding complexity.