On two high-rate algebraic space-time codes

We examine some algebraic properties of two high-rate linear space-time block codes over M=2,3 transmit antennas. Although these high-rate codes have positive coding gain, the gain decreases when increasing the constellation size. We give tight upper and lower bounds on the achieved coding gains as functions of the size of the constellations used. We show that when using the irrational numbers √3 and √2, the coding gains express the approximation of these numbers by continued fractions depending on the constellations used. The poor approximation of these numbers by rational numbers is then shown to make the coding gains decrease slowly when increasing the constellation size.