The BER performance of DRSC-SM scheme and reference schemes through quasi-static Rayleigh fading channel is presented in this section. Three cases of distrusted RS codes presented in Table 5 are mainly considered to generalize our proposed DRSC-SM scheme. For the first case, we employ the RS codes over GF(\(2^4\)) constructed based on the polynomial \(1+X+X^4\). Furthermore, the RS codes over GF(\(2^5\)) and GF(\(2^6\)) which are constructed using polynomials \(1+X^2+X^5\) and \(1+X+X^6\) are considered in the second and third cases, respectively. The modulation schemes such as 4-QAM, 8-QAM and 16-QAM are applied in the first, second and third cases, respectively. The SNR per bit between the \(S-D\) link is represented as \(\lambda _{S,D}\). Similarly, \(\lambda _{S,R}\) and \(\lambda _{R,D}\) denote the SNR per bit for the \(S-R\) and \(R-D\) links, respectively. It is assumed that the relay node has a 2 dB SNR gain over the source node, i.e., \(\lambda _{R,D}=\lambda _{S,D}+2\) dB. The Euclidean decoding algorithm is used for overall simulation. Furthermore, we assume that all corresponding receivers have perfect channel state information.

### BER performance of DRSC-SM scheme under different subcode design approaches

In this section, the BER performance of DRSC-SM scheme under different subcode design approaches for the first case is illustrated in Fig. 6. The ideal source-to-relay channel (\(\lambda _{S,R}=\infty\)), \(N_{\mathrm{{t}}}=N_{\mathrm{{r}}}=4\), maximum likelihood (ML) detection and joint RS decoding (smart algorithm) are used. The simulated results show the benefits of Approach 1 and Approach 2 as compared with the randomly selected approach, which is due to the fact that Approach 1 and Approach 2 effectively reduce the number of minimum weight codewords generated at the destination. The phenomenon illustrates the advantageous effect of proper symbol selection of our presented approaches on the performance of DRSC-SM scheme. In addition, Fig. 6 shows that the system performance under Approach 1 is very near to that under Approach 2 for low SNR. Also, for high SNR, the performance degradation is negligible. This is because the optimized code generated at the destination under the two search approaches have the same minimum distance (i.e., 3). Thus, this fully demonstrates the effectiveness of the local search Approach 2 over the brute-force search Approach 1. Furthermore, the system performance under (\(\lambda _{S,R}=\infty\) ) employing the smart decoding algorithm with the randomly selected approach and Approach 2 under ML detection and \(N_{\mathrm{{t}}}=N_{\mathrm{{r}}}=4\) for the second and third cases is presented in Figs. 7 and 8. It is seen that the DRSC-SM scheme with Approach 2 achieves better performance than that with the randomly selected approach. It can be explained that Approach 2 generates an optimized code with a larger minimum distance (i.e., 32 for the second case and 30 for the third case) at the destination.

Moreover, Figs. 6, 7 and 8 also show the system performance under the modified Approach 2 that has lower complexity than Approach 2. The simulation results demonstrate that the performance difference between the modified Approach 2 and Approach 2 is relatively small. For example, at SNR=14 dB in Fig. 6, the DRSC-SM scheme under Approach 2 and the modified Approach 2 achieves the approximate performances, i.e., \(3.2\times 10^{-6}\) and \(2.2\times 10^{-6}\). The phenomenon reflects the rationality of the modified Approach 2 with the reduced complexity.

### Performance of DRSC-SM scheme employing different decoding algorithms

The performance comparison of DRSC-SM employing joint decoding algorithms (naive algorithm, smart algorithm and improved smart algorithm) for the first case is shown in Fig. 9. The ideal source-to-relay channel condition, i.e., \(\lambda _{S,R}=\infty\), ML detection and \(N_{\mathrm{{t}}}=N_{\mathrm{{r}}}=4,\) are supposed. As shown in Fig. 9a, for SNR\(\ge 3\), the DRSC-SM system using the smart algorithm outperforms that using the naive one when the code resulted by Approach 1 is constructed at the destination. The reason behind the performance gains is that the more reliable decoded symbols of the \(RS_2\) decoder are used as the input to the \(RS_1\) decoder at SNR\(\ge 3\). However, in the case of SNR\(<3\), the smart algorithm lags behind the naive algorithm. The poor performance can be ameliorated by adopting the improved smart algorithm as illustrated in Fig. 9b, which is explained that the improved smart algorithm can flexibly select more reliable symbols based on the change of SNR so that an additional advantage is given to the improved smart DRSC-SM scheme. Furthermore, Figs. 10 and 11 analyze the BER performance under different decoding algorithms for the second and third cases with \(\lambda _{S,R}=\infty\), ML detection and \(N_{\mathrm{{t}}}=N_{\mathrm{{r}}}=4\). The simulation results confirm the effectiveness of our proposed improved smart algorithm again.

### BER performance of DRSC-SM scheme over non-ideal and ideal source-to-relay channels and noncooperative scheme

Next, the BER performance of DRSC-SM over more practical scenarios (\(\lambda _{S,R}\ne \infty\)) and noncooperative scheme for the first, second and third cases is analyzed as shown in Figs. 12, 13 and 14, respectively. Approach 1 is applied in the first case, but Approach 2 is applied in the other two cases. Furthermore, the joint decoding (smart decoding algorithm), \(N_{\mathrm{{t}}}=N_{\mathrm{{r}}}=4\) and ML detection are employed in the three cases. For a fair comparison, the DRSC-SM scheme and its corresponding noncooperative scheme have an identical code rate from the destination point of view. It is noticed from Fig. 12 that the DRSC-SM scheme under \(\lambda _{S,R}=\infty\) excels its noncooperative scheme under the same code rate, i.e., 13/30. At BER\(=3\times 10^{-5}\), the DRSC-SM scheme obtains a BER performance gain of approximately 2.7 dB over the noncooperative scheme. It can be explained that the cooperation gives the performance gains to the system. Furthermore, the performance of DRSC-SM scheme under \(\lambda _{S,R}=14\) dB is near to that of DRSC-SM scheme under \(\lambda _{S,R}=\infty\). However, the BER performance becomes worse when the SNR between the source *S* and relay *R* becomes poor, i.e., \(\lambda _{S,R}=8\) dB. For such a poor link, an error floor is exhibited at BER \(\approx 6\times 10^{-4}\). This is because the erroneous decoding at the relay node incurs the error propagation of destination node. The cyclic redundancy check (CRC) technique [9] can be applied to control the error propagation to some extent. However, the details are beyond the scope of this manuscript. Similarly, from Figs. 13 and 14, we not only discover the superiority of ideal DRSC-SM scheme (\(\lambda _{S,R}=\infty\)) over noncooperative scheme, but also observe the performance of DRSC-SM under \(\lambda _{S,R}=14\) dB and \(\lambda _{S,R}=19\) dB is near to that of ideal DRSC-SM scheme. However, when \(\lambda _{S,R}\) is taken as a poor value, i.e., 9 dB and 11 dB, the BER curves of Figs. 13 and 14 will be flat at high SNR regime.

### BER performance of DRSC-SM scheme with various number of receive antennas

Figures 15 and 16 discuss the BER performance of DRSC-SM scheme with various number of receive antennas for the second and third cases. In Monte Carlo simulations, the joint decoding (smart algorithm), Approach 2, ML detection and \(\lambda _{S,R}=\infty\) are assumed. The simulated results demonstrate that incresing \(N_{\mathrm{{r}}}\) improves the performance. In Fig. 15, at SNR=10 dB, the DRSC-SM scheme with \(N_{\mathrm{{r}}}=2\) receiving antennas gets BER= \(2.4\times 10^{-2}\). When the destination uses \(N_{\mathrm{{r}}}=3, 4\) and 6 receive antennas, the performances \(1.9\times 10^{-3}\), \(1.1\times 10^{-4}\) and \(2.4\times 10^{-7}\) are separately obtained at the same SNR. Furthermore, Fig. 16 presents that the DRSC-SM scheme under \(N_{\mathrm{{r}}}=2, 3, 4\) and 6 obtains BER= \(8.7\times 10^{-3}\), \(6.4\times 10^{-4}\), \(5\times 10^{-5}\) and \(4.1\times 10^{-7}\), respectively, at \(\lambda _{S,R}=13\) dB. This phenomenon can be explained that augmenting its value of \(N_{\mathrm{{r}}}\) adds the spatial diversity in DRSC-SM scheme that will eventually enhance the overall performance of the communication system.

### Performance of the proposed DRSC-SM scheme and the existing schemes

This section compares the performance of the proposed DRSC-SM scheme and the existing schemes, i.e., RS-coded cooperative SM (RSCC-SM) [16] and distributed RS coding (DRSC) [17] under \(\lambda _{S,R}=\infty\).

Figure 17 shows the simulation results of the proposed DRSC-SM scheme and the RSCC-SM scheme [16]. The simulation conditions such as \(N_{\mathrm{{t}}}=4\), 16-QAM, ML detection and smart decoding algorithm are used in the two schemes. Moreover, our proposed scheme adopts the optimized Approach 2 at the relay to get the optimized selection pattern by which partial symbols are selected from the source information symbols for further encoding. However, in the existing RSCC-SM scheme, the relay selects the partial symbols from the source information symbols for further encoding based on the random selection pattern. From Fig. 17, we observe that our proposed DRSC-SM outperforms the existing RSCC-SM. For example, at BER \(=1.4\times 10^{-4}\), the DRSC-SM scheme with \(N_{\mathrm{{r}}}=4\) is 0.9 dB better than the RSCC-SM with \(N_{\mathrm{{r}}}=4\). At the same BER, under \(N_{\mathrm{{r}}}=6\), the DRSC-SM scheme is 1 dB better than its counterpart. The main reason for such an attractive performance gain is that our proposed DRSC-SM scheme enables to construct a code with a larger minimum distance (i.e., 48) at the destination by appropriately selecting the partial information symbols at the relay.

Figure 18 shows the performance comparison between the proposed DRSC-SM scheme and the existing DRSC scheme [17]. The same conditions such as \(RS_1(31,27)\), \(RS_2(31,17)\), Approach 2, ML detection and smart decoding algorithm are adopted in the two compared schemes. The simulated results in Fig. 18 reflect the superiority of our proposed scheme. For instance, at SNR \(=13\) dB, the DRSC-SM scheme achieves a very promising performance (i.e., \(4.3\times 10^{-6}\)) but the DRSC scheme obtains a poor performance (i.e., \(2.6\times 10^{-3}\)). The performance enhancement is mainly because the adopted novel SM technique provides spatial diversity for the proposed DRSC-SM scheme.

### Performance comparison between DRSC-SM and DRSC-V-BLAST schemes

In Sect. 6.5, Fig. 18 exhibits the performance advantages of our proposed DRSC-SM scheme over the DRSC (without SM). To further confirm the performance superiority of our proposed DRSC-SM scheme, we compare it with the distributed RS-coded V-BLAST (DRSC-V-BLAST) scheme under an identical spectral efficiency. The condition \(\lambda _{S,R}=\infty\) is supposed. In our proposed DRSC-SM scheme, \(N_{\mathrm{{t}}}=4\) and 4-QAM are used. However, \(N_{\mathrm{{t}}}=2\) and 4-QAM are used in the DRSC-V-BLAST scheme. At the destination, the smart decoding algorithm is used to jointly recover the source information. Moreover, the DRSC-SM scheme utilizes the maximum ratio combining (MRC) reception [20], and the minimum mean squared error (MMSE) detection [20] is adopted in the DRSC-V-BLAST scheme. The significant performance gains of the DRSC-SM scheme over the DRSC-V-BLAST scheme are shown in Fig. 19. For instance, under \(N_{\mathrm{{r}}}=4\) and 6, 0.5 dB and 0.4 dB gains are obtained by our system at BER=\(10^{-4}\). The main reason behind the attractive gains is that our presented DRSC-SM scheme can completely avoid the drawback (ICI) of the DRSC-V-BLAST scheme so that the ability to recover the source information is improved.