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It has been shown that when full channel state information (CSI) is available at the transmitter, the performance of space-time coded system can be further improved. In this paper, we present the design of combined space time trellis codes and beamforming, henceforth referred to as weighted space time trellis codes (WSTTCs). The upper bound on the pairwise error probability (PEP) of WSTTCs is derived, and a new design criterion is proposed. It is shown that WSTTCs can achieve not only a full spatial diversity order, but also an additional time diversity order as well as an additional received SNR gain, referred to as the weighting gain and a significant coding gain, relative the standard STTC. A good 4-state code based on the proposed criterion has been found by computer search. Simulation results show that the proposed 4-state WSTTC not only outperforms the optimum 4-state STTC without weighting by more than 4 dB, but is also superior by about 4 dB to the conventional beamforming scheme at the BER of 1.0e-4, over fast fading channels.