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In this paper, we consider the problem of space-time (ST) coding with pulse position modulation (PPM). While all the existing ST block codes necessitate rotating the phase or amplifying the amplitude of the transmitted symbols, the proposed scheme can be associated with unipolar PPM constellations without introducing any additional constellation extension. In other words, full transmit diversity can be achieved while conveying the information only through the time delays of the modulated signals transmitted from the different antennas. The absence of phase rotations renders the proposed scheme convenient for low- cost carrier-less multiple-input-multiple-output (MIMO) time- hopping ultra-wideband (TH-UWB) systems and for MIMO free-space optical (FSO) communications with direct detection. In particular, we propose two families of minimal-delay ST block codes that achieve a full transmit diversity order with PPM. Designate by n the number of transmit antennas and by M the number of modulation positions. For a given set of values of (n, m), the first family of codes achieves a rate of 1 symbol per channel use (PCU) which is the highest possible achievable rate when no constellation extensions are introduced. The second family of codes can be applied with a wider range of (n, m) at the expense of a reduced rate given by: R=1/n+n-1/n log2(M-1)/n log2(M).