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The design requirements for space-time coding typically involves achieving the goals of good performance, high rates, and low decoding complexity. In this paper, we introduce a further constraint on space-time code design in that the code should also lead to low values of the peak-to-mean envelope power ratio (PMEPR) for each antenna. Towards that end, we propose a new class of space-time codes called the "low PMEPR space-time" (LPST) codes. The LPST codes are obtained using the properties of certain cyclotomic number fields. The LPST codes achieve a performance identical to that of the threaded algebraic space-time (TAST) codes but at a much smaller PMEPR. With M antennas and a rate of one symbol per channel use, the LPST codes lead to a decrease in PMEPR by at least a factor of M relative to a Hadamard spread version of the TAST code. For rates beyond one symbol per channel use and up to a guaranteed amount, the LPST codes have provably smaller PMEPR than the corresponding TAST codes. Additionally, with the concept of punctured LPST codes proposed in this paper, significant performance improvement is obtained over the full diversity TAST schemes of comparable complexity. Numerical examples are provided to illustrate the advantage of the proposed codes in terms of PMEPR reduction and performance improvement for very high rate wireless communications.