Gaussian integers and interleaved rank codes for space–time block codes

Abstract

This paper presents the design of space-time block codes (STBCs) over maximum rank distance (MRD) codes, energy-efficient STBCs, STBCs using interleaved-MRD codes, the use of Gaussian integers for STBCs modulation, and Gabidulin's decoding algorithm for decoding STBCs. The design fundamentals of STBCs using MRD codes are firstly put forward for different number of transmit antennas. Extension finite fields (Galois fields) are used to design these linear block codes. Afterward, a comparative study of MRDbased STBCs with corresponding orthogonal and quasi-orthogonal codes is also included in the paper. The simulation results show that rank codes, for any number of transmit antennas, exhibit diversity gain at full rate contrary to orthogonal codes, which give diversity gain at full rate only for two transmit antennas case. Secondly, an energy-efficient MRD-STBC is proposed, which outperforms orthogonal STBC at least for 2 1 antenna system. Thirdly, interleaved-MRD codes are used to construct higher-order transmit antenna systems. Using interleaved-MRD codes further reduces the complexity (compared with normal MRD codes) of the decoding algorithm. Fourthly, the use of Gaussian integers is utilized in mapping MRD-based STBCs to complex constellations. Furthermore, it is described how an efficient and computationally less complex Gabidulin's decoding algorithm can be exploited for decoding complex MRD-STBCs. The decoding results have been compared against hard-decision maximum likelihood decoding. Under this decoding scheme, MRD-STBCs have been shown to be potential candidate for higher transmit antenna systems as the decoding complexity of Gabidulin's algorithm is far less, and its performance for decoding MRD-STBCs is somewhat reasonable.In this section, the design criteria of STBC and fundamentals of MRD codes are described. Design criteria for STBCIt is useful to present the information theory fundamentals behind the construction of STBCs. It is necessary to highlight that the following calculations are valid for Rayleigh fading channels. These vary for Rician and other channels [19]. However, in this paper, our discussion is limited to mere Rayleigh channels only; therefore, this fact will not disturb the results presented in this paper in any way.We start by assuming the transmission matrix as X i ; then Q X is the corresponding estimated matrix, which was obtained by ML decoding. Next, the probability of error (P(e)), that is,can be calculated based on Euclidean distance, d Eu as follows [20]: Int. J. Commun. Syst. 2017; 30: e2943 d M .x;´/ > d M .x; y/ C d M .y;´/ Namely, let x D u C .m C s/i; y D i;´D 0. By direct calculation, one can show that x D x.Hence, all three x,y,and´are in MZ . We have Energy-efficient MIMO systems are obtaining the attention of researchers recently in many aspects [27]. In this section, it is described how energy-constraint STBCs are formed using both MRD and orthogonal codes. We can make use of a specific primitive irreducible polynomial over GF.2/ to form our desired codes. Selecti...

Design of MIMO-Visible Light Communication Transceiver Using Maximum Rank Distance Codes

“…A linear code can be represented as C = (n, k, d), where n relates to the length of the code, k indicates the quantity of symbols and d signifies the code distance. MRD codes are based on metric 'rank' constructed by Gabidulin [43] and Gabidulin and Pilipchuk [44] that attempts to detect and correct more number of errors in comparison with other codes based on metric 'minimum hamming distance' [45]. This error handling capability is due to linear dependency among different symbols.…”