Space-time group codes

We consider the design of space-time constellations based on group codes for fading channels with multiple transmit and receive antennas. These codes can be viewed as multi-antenna extensions of phase-shift keying, in the sense that all codewords have equal energy, all are rotations of a fixed codeword, and there is a simple differential transmission rule that allows data to be sent without channel estimates at the transmitter or receiver. For coherent detection, we show that all optimal full-rank space-time group codes are unitary (each code matrix has equal-energy, orthogonal rows). This leads to a simpler code design criterion and shows that unitary codes play an important role in coherent as well as noncoherent communication. For any number of transmit antennas t, we use the design criterion to characterize all full-rank unitary group codes of minimum blocklength (also t) with 2/sup p/ codewords. In principle, these results allow us to determine all optimal unitary group codes with square code matrices, which matches the class of codes proposed recently for differential modulation.