Optimization of full-rate full-diversity linear space-time codes using the union bound

Although many space-time codes sacrifice their rate in order to achieve a high diversity order, such a sacrifice is not necessary. Recent work has reported two instances of a linear space-time code that achieves both a full rate of min(t, r) and a full diversity order of tr over a t-input r-output Rayleigh-fading channel (Damen, M.O. et al., IEEE Trans. on Inf. Theory, vol.48, no.3, p.753-60, 2002; Ma, X. and Giannakis, G.B., Proc. 2nd Sensor Array and Multichannel SP Workshop, p.442-6, 2002). We show that such full-rate full-diversity codes are plentiful and can, in fact, be found with probability one by randomly choosing an encoding matrix from an ensemble of matrices with orthonormal columns. However, full rate and full diversity does not guarantee good error-rate performance. Different encoding matrices with the same rate and diversity order can have markedly different error rates. We propose the union bound on word-error rate as an optimization metric and perform constrained optimization to find good space-time codes. For the two-input, two-output Rayleigh channel, we present an optimized code that outperforms the previously reported codes by 1.25 dB at 4 b/s/Hz and a frame-error rate of 10^-3.