The golden code: a 2 x 2 full-rate space-time code with non-vanishing determinants

Abstract

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.

Cyclic Division Algebras: A Tool for Space-Time Coding

“…In [7,44] and then in [21], perfect codes were presented as division algebra codes which furthermore satisfy a shaping property and have a non-vanishing determinant. In [53], information lossless codes from crossed product algebras, a new family of division algebras, are presented.…”

“…In parallel, in [7,15,33,63], the first 2 × 2 codes achieving the diversity-multiplexing gain trade-off of Zheng and Tse [64] were found. It was furthermore shown [63] that a necessary condition to achieve the trade-off for a 2 × 2 code is actually to have a non-vanishing determinant (though not stated with this terminology).…”

“…It was furthermore shown [63] that a necessary condition to achieve the trade-off for a 2 × 2 code is actually to have a non-vanishing determinant (though not stated with this terminology). In [7], it was shown that the algebraic structure of cyclic division algebras was the key for constructing 2 × 2 non-vanishing determinant codes. In [20], it was shown more generally that division algebra codes are a class of codes that achieve the trade-off, thanks to the non-vanishing determinant.…”

Performance analysis of multi‐antenna selection policies using the golden code in multiple‐input multiple‐output systems

“…, and β is set as √ −1, |µ| is set to unity to satisfy the non-vanishing determinant and ensures the same average power is transmitted [11]. The information bits are respectively encoded with the Golden Code matrix (1).…”

“…The Golden Code matrix is a full-rate, full diversity and full rank linear 2 × 2 linear dispersion algebraic space-time code for two transmit antennas and two or more receive antenna MIMO system [11]. We focus on selecting the best two transmit antennas from N participating antenna by using maximum-minimum and maximum-sum selection strategies.…”