Optimal space-time constellations from groups

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 multiantenna extensions of phase-shift keying (PSK), 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 suggests that unitary codes may play an important role in coherent as well as noncoherent communication. For any number of transmit antennas t, we then use the design criterion to characterize all full-rank unitary space-time group codes of minimum block length (also t) which have 2^p codewords. These results allow us to characterize all optimal 2^p-ary unitary group codes with square code matrices. This restricted class of block codes matches the class proposed for differential modulation by Hughes (see IEEE Trans. Inform. Theory, vol.46, p.2567-78, Nov. 2000), and by Hochwald and Sweldens (see IEEE Trans. Commun., vol.48, p.2041-2052, Dec. 2000).