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Abstract—In this correspondence, we propose some new designs of P P unitary space–time codes of sizes 6, 32, 48, 64 with best-known diversity products (or product distances) by partially using sphere packing theory. In particular, we present an optimal P P unitary space–time code of size T in the sense that it reaches the maximal possible diversity product for P P unitary space–time codes of size T. The construction and the optimality of the code of size T provide the precise value of the maximal diversity product of a P P unitary space–time code of size T. Index Terms—Differential space–time modulation, optimal diversity product, packing theory, unitary space–time codes. I.

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, where the demodulation is in fact the same as the coherent demodulation of space–time block coding by replacing the channel matrix with the previously received signal matrix. 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 in joint with a conventional binary code and joint iterative decoding/demodulation with a superior performance. 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). When a USTC for the DSTM forms a

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