We construct a full data rate space-time (ST) block code over = 2 transmit antennas and = 2 symbol periods, and we prove that it achieves a transmit diversity of 2 over all constellations carved from [ ]. Further, we optimize the coding gain of the proposed code and then compare it to the Alamouti code. It is shown that the new code outperforms the Alamouti code at low and high signal-to-noise ratio (SNR) when the number of receive antennas 1. The performance improvement is further enhanced when or the size of the constellation increases. We relate the problem of ST diversity gain to algebraic number theory, and the coding gain optimization to the theory of simultaneous Diophantine approximation in the geometry of numbers. We find that the coding gain optimization is equivalent to finding irrational numbers "the furthest" from any simultaneous rational approximations.

We explore in this paper the lattice sphere packing representation of a multi-antenna system and the algebraic space-time (ST) codes. We apply the sphere decoding (SD) algorithm to the resulted lattice code. For the uncoded system, SD yields, with small increase in complexity, a huge improvement over the well-known V-BLAST detection algorithm. SD of algebraic ST codes exploits the full diversity of the coded multi-antenna system, and makes the proposed scheme very appealing to take advantage of the richness of the multi-antenna environment. The fact that the SD does not depend on the constellation size, gives rise to systems with very high spectral efficiency, maximum likelihood (ML) performance, and low decoding complexity. I . INTRODUCTIONRecently, the field of multi-antenna processing and space-time (ST) coding has attracted large interest in the communication community due to the huge capacity of the multi-antenna environment [l]. Because of the maximum likelihood (ML) detection high complexity sub-optimal detection like the V-BLAST have been proposed for the uncoded system [2].In this paper, we prove that one can reach the ML performance of the uncoded system with low complexity by applying the sphere decoder [3] on the lattice sphere packing representation of a multi-antenna system. Moreover, it is shown that one can achieve the full diversity of the multi-antenna system by using full spatial diversity rotated constellations without adding redundancy [4], and still reach the ML performance with reasonable complexity. SIMULATION RESULTSIn simulations we use the constellation q-QAM, with q = 4 , 16. The average energy per bit is fixed to E b = 1. We consider a multi-antenna system with M transmitters and N = M receivers. The algebraic coding is done over 1 periods by using rotated constellations of dimension MZ. The channel transfer matrix is modeled by independent complex Gaussian random variables of variance 0.5 per real dimension. The curves are plotted as a function of SNR (the signal-to-noise ratio per bit), and the variance g 2 of the complex AWGN per real dimension is adjusted by the formula m2 = ,zf;p) 10-SNR''o , where E, is the average symbol energy of the q-QAM when E b = 1 and equals v. In figures 1 and 2 we applied the SD on both uncoded data streams and algebraic ST codes over 1 periods with M = N transmit/receive antennas [4]. Comparisons are done with the V-BLAST detection algorithm [2]. It is shown that at the expense of a moderate increase in complexity, a huge improvement in performance is achieved. 10 E c B 10' B 2 10' 10 1 1 12 13 14 1s 16 17 in SNR per D 1 Fig. 1: SD of V-BLAST architecture, M = N = 8 , averagesymbol error rate of the 16-QAM modulation, 32 bits/s/Hz. b 0 2 4 6 6 10 12 14 18 18 SNR per bn 10 Fig. 2: SD of algebraic ST codes, M = N = 4, average symbol error rate of the 4-QAM modulation, 8 bits/s/Hz, 1 = 1 , 2 , 3 . REFERENCES V. Tarokh, N. Seshadri and A. Calderbank, "Space-time codes for high data rate wireless communications: performance criterion and code construction,"...

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