We construct a class of linear quasi-orthogonal space-time block codes that achieve full diversity over quasistatic fading channels for any transmit antennas. These codes achieve a normalized rate of one symbol per channel use. Constellation rotation is shown to be necessary for the full-diversity feature of these codes. When the number of transmit antennas is a power of 2, these codes are also delay “optimal.” The quasi-orthogonal property of the code makes one half of the symbols orthogonal to the other half, and we show that this allows each half to be decoded separately without any loss of performance. We give an iterative construction of these codes with a practical decoding algorithm. Numerical simulations are presented to evaluate the performance of these codes in terms of capacity as well as probability of error versus SNR curves. For some special cases, we compute the pairwise probability of error averaged over all the channel states as a single integral that shows the diversity and coding gain more clearly.