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Differential space-time modulation (DSTM) has been recently proposed by Hughes, and Hochwald and Sweldens when the channel information is not known at the receiver. On the other hand, the DSTM also needs a recursive memory of a matrix block at the encoder and therefore provides a trellis structure when the channel information is known at the receiver, which is the interest of this paper. This recursive structure of the DSTM has been adopted lately by Schlegel and Grant as an inner trellis code concatenated with an outer binary code to achieve turbo gain. The number of states of the trellis from the recursive structure depends on both the memory size, which is fixed in this case, and the unitary space-time code (USTC). In this paper, we propose a new USTC design criterion to ensure that the trellis structure improves the diversity product over the USTC as a block code. Based on the new criterion, we propose a new class of USTC design for an arbitrary number of transmit antennas that has an analytical diversity product formula for two transmit antennas. We then follow Schlegel and Grant's approach for joint encoding and iterative decoding of a binary coded DSTM (turbo space-time coding) and numerically show that our new USTC designs for the recursive space-time trellis modulation outperforms the group USTC used by Schlegel and Grant.