Generalized Silver Codes

For an $nt$ transmit, $nr$ receive antenna system ($nt \times nr$ system), a {\it{full-rate}} space time block code (STBC) transmits $n{min} = min(nt,nr)$ complex symbols per channel use. The well known Golden code is an example of a full-rate, full-diversity STBC for 2 transmit antennas. Its ML-decoding complexity is of the order of $M^{2.5}$ for square $M$-QAM. The Silver code for 2 transmit antennas has all the desirable properties of the Golden code except its coding gain, but offers lower ML-decoding complexity of the order of $M^2$. Importantly, the slight loss in coding gain is negligible compared to the advantage it offers in terms of lowering the ML-decoding complexity. For higher number of transmit antennas, the best known codes are the Perfect codes, which are full-rate, full-diversity, information lossless codes (for $nr \geq nt$) but have a high ML-decoding complexity of the order of $M^{ntn{min}}$ (for $nr < nt$, the punctured Perfect codes are considered). In this paper, a scheme to obtain full-rate STBCs for $2^a$ transmit antennas and any $nr$ with reduced ML-decoding complexity of the order of $M^{{nt(n{min}-(3/4))-0.5}$,} is presented. The codes constructed are also information lossless for $nr \geq nt$, like the Perfect codes and allow higher mutual information than the comparable punctured Perfect codes for $nr < nt$. These codes are referred to as the {\it generalized Silver codes}, since they enjoy the same desirable properties as the comparable Perfect codes (except possibly the coding gain) with lower ML-decoding complexity, analogous to the Silver-Golden codes for 2 transmit antennas. Simulation results of the symbol error rates for 4 and 8 transmit antennas show that the generalized Silver codes match the punctured Perfect codes in error performance while offering lower ML-decoding complexity.