## 3 Results and Discussion
For the MUSA\({}_{0}\) set, 25 codes with low correlation properties, in ascending order of their intercorrelation value, selected from the MUSA set previously defined, and the rest of the 7 codes have been replaced with ones with zero cross-correlation, also from the matrix given in Equation (16). Then, the extended MUSA (EMUSA) set of codes was created by extending the MUSA and MUSA\({}_{0}\) from length 8 to length 32, obtaining thus EMUSA and EMUSA\({}_{0}\) sets. The results in terms of BER versus SNR in the same configuration as the one previously used and with (\(1\times 30\times 30\)) antennas are shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c} \hline  & **(1 \(\times\) 26 \(\times\) 30)** & **(1 \(\times\) 30 \(\times\) 30)** & **(1 \(\times\) 26 \(\times\) 30)** & **(1 \(\times\) 30 \(\times\) 30)** \\  & **K = 12** & **K = 12** & **K = 16** & **K = 16** \\ \hline Walsh & 0.0006593 & 0.0001083 & 0.004657 & 0.001846 \\ PN & 0.001593 & 0.0005998 & 0.006406 & 0.00559 \\ \hline \end{tabular}
\end{table} 
Table 2: BER vs. SNR for Walsh and PN spreading codes for SNR = 12 dB. Figure 4: BER vs. SNR for a (\(1\times 26\times 30\)) and (\(1\times 30\times 30\)) MIMO configurations, QPSK modulation, PN code with 12, 16 active users. Figure 5: BER vs. SNR for a (\(1\times 30\times 30\)) MIMO configuration, QPSK modulation, MUSA and EMUSA complex spreading codes with 16 active users. If the complex spreading codes are obtained starting from PN real spreading codes taking into consideration the \(g^{6}\ +\ g\ +\ 1\) generating polynomial, where \(g\) is the delay element, following the principles stated in Section 2.3, Figure 8 is obtained. It was considered the most unfavorable situation obtained in the situations above, namely (\(1\times 26\times 30\)), when the number of active users was equal to 16. Also, simulations were made in (\(1\times 10\times 26\)) and (\(1\times 10\times 30\)) configurations, but the BER performance was very low. Thus, a question has arisen: Maintaining the length of the code at 63, how much can we reduce the number of the antennas at the receiver, and therefore the system hardware complexity, such that the performances should be still acceptable (the BER should be below a given threshold)? Thus, the number of antennas at the relay and at the BS was decreased to 10, 8, 4, and then 2. Setting a threshold of \(10^{-4}\) for the BER, in (\(1\times 10\times 10\)) situation, it was reached at SNR = 8 dB, at SNR = 10 dB in the (\(1\times 8\times 8\)), and at SNR = 13 dB in the (\(1\times 4\times 4\)) situation. This shows that, using these complex PN codes, the number of antennas can be reduced significantly and, therefore, the associated costs. In order to compare the efficiency of the present approach, we tried to compare our results with similar ones in the existing literature. Thus, in [39], the authors have obtained several results in somewhat similar conditions to our approach but using an MMSE-SIC detector at the receiver and with different numbers of antennas at the receiver and transmitter, an approximate value for BER of \(5\times 10^{-2}\) was obtained for SNR, for 12 dB, while with our setup, we obtained BER less than \(10^{-3}\) in all configurations. In [40], the authors obtained an approximate value for BER of \(10^{-4}\) at a SNR of 12 dB, so in this case, the results were slightly better than MUSA but lower than MUSA\({}_{0}\). In [43], a similar setup has been used, with \(\
icefrac{{1}}{{4}}\) LDPC codes and QPSK/OFDM modulation, but the codes were the ones used in the 3 GPP standard. The number of antennas was (\(1\times 2\)), and there was no relay between source and destination, and the channel was affected by AWGN only, obtaining a BER of approximately \(5\times 10^{-2}\) at SNR = 15 dB for MUSA, 12 users, and of about \(5\times 10^{-3}\) for overloaded SDMA-OFDMA-MLD. Figure 8: BER vs. SNR for a (\(1\times 2\times 2\)), (\(1\times 4\times 4\)), (\(1\times 8\times 8\)), and (\(1\times 10\times 10\)) MIMO configuration, QPSK modulation, cPN with 16 active users.