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Space-time trellis codes (STTC), that provide transmit diversity and coding gain, have been known to give good performance in fading channels. So far, most work on STTC has focused on full rate STTC. However, given the transmission rate of the STTC its performance can be improved by using an expanded signal set. In this paper we obtain the optimal STTC on an expanded signal set in terms of a measure of coding gain.