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A family of space-time codes suited for noncoherent multi-input multi-output (MIMO) systems is presented. These codes use all the complex degrees of freedom of the system, i.e. M/spl times/(1-(M/T)) symbols per channel use. They are constructed as codes on the Grassmann manifold G/sub T,M/(/spl Copf/) where T is the temporal codelength and M is the number of transmit antennas.