## 4 Simulation Settings and Results
### Simulation Results for the Multi-Way Relaying System
It is to be noted that the multi-way relaying system employing the \((512,416)\) code can support 416 users, and the one employing the \((128,104)\) code can support 104 users. This indicates the ability of the multi-way relaying system using polar codes to support a large number of users, similar to the system in ref. [12] that uses LDPC codes. The BER performance of a \((512,416)\) polar code with SC and SCL decoding is illustrated in Figure 8 and Figure 9, respectively. Compared to the \((8,5)\) polar code, the \((512,416)\) polar code provides a more significant performance gain with both decoding methods over the uncoded scheme in all cases. The SCL decoding method has a better performance of about 2 dB to achieve a BER of \(10^{-6}\) in all four cases compared to SC decoding. The results when the thresholding technique is employed at user terminals are also included in Figures 8 and 9. It is seen that with the thresholding technique, the performance loss with respect to case 2 is reduced from 4 dB to about 1.7 B at a BER of \(10^{-6}\), demonstrating the effectiveness of this practical approach. The required \(E_{b}/N_{0}\) to achieve a BER of \(10^{-6}\) for the \((512,416)\) polar code is presented in Table 7. This table also presents similar results for various LDPC codes employing log-BP decoding from [12]. The \((437,361)\) LDPC code is constructed using the method given in ref. [63]. All the codes have comparable codeword lengths and rates equal or higher than 0.8. In the ideal case when all the relays transmit, the \((512,416)\) polar code with SCL decoding can achieve a better performance than the two LDPC codes with log-BP decoding. This illustrates the effectiveness of polar codes with SCL decoding for our system when all relays transmit. In the other three non-ideal cases, the polar code with SCL decoding also achieves a performance gain of about 2 dB compared to the \((480,400)\) LDPC code. In these cases, the BER performance with SC decoding is better than or comparable to the \((480,400)\) LDPC code. However, the polar code with SCL decoding has a performance loss of about 1.6 dB compared to the \((437,361)\) LDPC code in these three cases. This is because  
Figure 7: Bit error rate (BER) performance of three long polar codes when all relays transmit. the relays of the polar code system have larger relevant group sizes. The average relevant group size of the relays in the \((512,416)\) polar code system is \(69\), while the average group size is only \(19\) for the \((437,361)\) LDPC code system. With a larger relevant group, for the same \(E_{b}/N_{0}\), the relays have a higher probability of detecting a wrong symbol and hence remain silent. This explains the performance degradation of polar codes in non-ideal cases when compared to LDPC codes with a smaller average group size. Although there is a small performance gap between the polar codes and some LDPC codes, the low decoding complexity analyzed in Section 3 makes polar codes competitive for application in our system. A specific construction method for polar codes that also takes into account the relays' average group sizes could result in even better performance. It was observed in [35] that systematic polar codes have a better BER performance compared to non-systematic polar codes. To test the effects of systematic polar encoding in our multi-way relaying system, we first generate the information bits of polar codes \(\mathbf{m}_{A}\), then the terminal bits \(\mathbf{m}_{T}\) are obtained from (36). The BER performance of \(\hat{\mathbf{m}}_{A}\) and \(\hat{\mathbf{m}}_{T}\) for the \((512,416)\) polar code with SC decoding is presented in Figure 10, and the performance with SCL decoding is presented in Figure 11. The curves corresponding to \(\hat{\mathbf{m}}_{T}\) and \(\hat{\mathbf{m}}_{A}\) are marked with \"Ter.\" and \"polar\", respectively. In all cases, \(\hat{\mathbf{m}}_{T}\) results in a better error performance compared to \(\hat{\mathbf{m}}_{A}\). The performance advantage of using systematic codes over