In this paper we present the Golden code for a 2 × 2 MIMO system. This is a full-rate 2 × 2 linear dispersion algebraic space-time code with unprecedented performance based on the Golden number 1+√ 5 2 . Full rate and full diversity algebraic codes for 2 × 2 coherent MIMO systems, were first constructed in [2], using number theoretical methods. This approach was later generalized for any number of transmit antennas M [5,4]. The above constructions satisfy the rank criterion and attempt to maximize, for a fixed signal set S, the coding advantage. A general family of 2 × 2 full-rank and full-rate linear dispersion space-time block codes (LD-STBC), based on quaternion algebras, was given in [1,6].Let K = Q(θ) be a quadratic extension of Q(i), we define the infinite code C∞ as the set of matrices of the formC∞ is clearly a linear code, i.e., X1 +X2 ∈ C∞ for all X1, X2 ∈ C∞. The finite code C is obtained by limiting the information symbols to a, b, c, d ∈ S ⊂ Z[i], where we assume the signal constellation S to be a 2 b -QAM, with in-phase and quadrature components equal to ±1, ±3, . . . and b bits per symbol. The code C∞ is a discrete subset of a cyclic division algebra over Q(i), obtained by selecting γ = N K/Q(i) (x) for any x ∈ K [1, 6]. A division algebra naturally yields a structured set of invertible matrices that can be used to construct square LD-STBC since for any codeword X ∈ C∞ the rank criterion is satisfied as det (X) = 0.We define the minimum determinant of C∞ asand the minimum determinant of the finite code C asMinimum determinants of C∞ in all previous constructions [2, 5, 4, 6] are non-zero, but vanish as the spectral efficiency b of the signal constellation S is increased. This problem appears because either transcendental elements or algebraic elements with a too high degree are used to construct the division algebras. Non-vanishing determinants may be of interest, whenever we want to apply some outer block coded modulation scheme, which usually entails a signal set expansion, if the spectral efficiency has to be preserved.In order to obtain energy efficient codes we need to construct a lattice M Z[i] 2 , a rotated version of the complex lattice 1 This work was supported in part by CERCOM. 2 4 6 8 1 0 1 2 1 4 16 18 20 22 24 26 Eb/N0 (dB) 1e-06 1e-05 0,0001 0,001 0,01 0,1 1 Bit Error Rate Cg, 16-QAM Cg, 4QAM Cb16, 16 QAM Cb4, 4 QAM Figure 1: Performance comparison of the new codes vs. those of [2] and [3]where M is a complex unitary matrix, so that there is no shaping loss in the signal constellation emitted by the transmit antennas. This additional property was never considered before and is the key to the improved performance our codes.The algebraic construction yields codewords of the Golden code of the form 1 √ 5where α = 1+i(1−θ), θ = 1+ √ 5 2 andθ = 1− √ 5 2 . We show that the Golden code has non-vanishing δmin(C∞) = 1/5, hence δmin(C) = 1/5 for any size of the signal constellation. Fig. 1 shows how the Golden code outperforms all previous constructions.

Abstract-In this paper, we introduce the notion of perfect space-time block codes (STBCs). These codes have full-rate, full-diversity, nonvanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. We present algebraic constructions of perfect STBCs for 2, 3, 4, and 6 antennas.

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The problem of designing Space-Time codes on the MIMO quasi-static channel have received considerable attention these last years. We now know how to design Perfect SpaceTime Block Codes, that is linear information preserving codes achieving the Diversity-Multiplexing gain (D-M) tradeoff [1], [2]. Recent standards using multiple antennas terminals such as IEEE 802.11n or IEEE 802.16e, for example, are based on OFDM. By using interleaving, such OFDM systems can be seen as parallel MIMO quasi-static channels. We propose, here, new perfect space-time block codes for parallel MIMO channels.