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-A key issue in compute-and-forward for physical layer network coding scheme is to determine a good function of the received messages to be reliably estimated at the relay nodes. We show that this optimization problem can be viewed as the problem of finding the closest point of Z [i] n to a line in the n-dimensional complex Euclidean space, within a bounded region around the origin. We then use the complex version of the LLL lattice basis reduction (CLLL) algorithm to provide a reduced complexity suboptimal solution as well as an upper bound to the minimum distance of the lattice point from the line. Using this bound we are able to find a lower bound to the ergodic rate and a union bound estimate on the error performance of a lattice constellation used for lattice network coding. We compare performance of the CLLL with a more complex iterative optimization method as well as with a simple quantized search. Simulations show how CLLL can trade some performance for a lower complexity.Index Terms-Ergodic rate, compute-and-forward, CLLL algorithm, quantized error, successive refinement.

Multiple antennas at both the transmitter and receiver ends of a wireless digital transmission channel may increase both data rate and reliability. Reliable high rate transmission over such channels can only be achieved through Space-Time coding. Rank and determinant code design criteria have been proposed to enhance diversity and coding gain. The special case of full-diversity criterion requires that the difference of any two distinct codewords has full rank.Extensive work has been done on Space-Time coding, aiming at finding fully diverse codes with high rate. Division algebras have been proposed as a new tool for constructing Space-Time codes, since they are non-commutative algebras that naturally yield linear fully diverse codes. Their algebraic properties can thus be further exploited to improve the design of good codes.The aim of this work is to provide a tutorial introduction to the algebraic tools involved in the design of codes based on cyclic division algebras. The different design criteria involved will be illustrated, including the constellation shaping, the information lossless property, the non-vanishing determinant property, and the diversity multiplexing trade-off. The final target is to give the complete mathematical background underlying the construction of the Golden code and the other Perfect Space-Time block codes.

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