## 4 Simulation Settings and Results
### Simulation Results for the Multi-Way Relaying System  
We consider four different cases based on the terminals' knowledge of the relays' transmission: (a) the ideal case when all relays transmit; (b) the case when the terminals do not know which relay transmits (case 1); (c) the case when the terminals know which relay transmits (case 2); (d) the case when the terminals do not know which relay transmits, but they use the hard threshold method. In (a), all the terminal-relay transmissions are perfect, and the relays detect all the signals correctly and hence transmit continuously. In (b), although some relays remain silent, the terminals assume the relays always transmit, and all channel LLRs are calculated using (40). In (c), the terminals know which relays transmit and modify all the channel LLRs corresponding to the relays that do not transmit based on (45). In (d), the terminals employ the hard threshold technique to modify the channel LLRs corresponding to the relay signals that do not pass the threshold. Figure 5: Frame error rate (FER) performance of a \((1024,512)\) polar code with successive cancellation (SC) and successive cancellation list (SCL) decoders over an additive white Gaussian noise (AWGN) channel and comparison with the results from ref. [60] (marked as [ref]). In this work, we consider equal power allocation to all nodes and hence assume unity transmission power \(P=1\) in all simulations. For each bit error rate (BER) point, we simulate at least 1000 randomly generated codewords with independent channel fading realizations and collect at least 500 bit errors. The list size for SCL decoding is set to \(L_{max}=8\). The bit energy \(E_{b}\) is related to the transmission power through the following:  
\[E_{b}=P\cdot\frac{N}{K} \tag{50}\]  
The BER performance of a system with five terminals and three relays employing a \((8,5)\) polar code with SC decoding is shown in Figure 6. The performance results for uncoded DBPSK transmission over a Rayleigh fading channel and a polar code with ML decoding are also included as references. We see that SC decoding achieves the same diversity order as ML decoding. When all the relays transmit, the error performance of SC decoding is very close to the performance of ML decoding. In the non-ideal case 1, we see that SC decoding provides a small advantage over ML decoding, showing that it enjoys a robustness to intermittent relay transmissions. At a BER of \(10^{-5}\), the performance gap between SC and ML decoding in case 1 is about 0.8 dB. For case 2, when the terminals know which relays transmit, the performance of ML and SC decoding becomes essentially the same. In summary, the error performance of polar codes with SC decoding is very close to ML decoding for short polar codes. The performance with SCL decoding for our multi-way relaying system with such short polar codes is essentially the same as with SC decoding; hence, we do not include these results. We will see that with longer polar codes, SCL decoding provides performance advantages over SC decoding. The BER performance of three longer polar codes with a rate of 0.8125, when used in our multi-way relaying system with all relays transmitting, is shown in Figure 7. All the codes have a significant performance gain compared to the uncoded scheme. At a BER of \(10^{-5}\), with SC decoding, the \((512,416)\) code has a performance gain of 2.5 dB compared to the \((128,104)\) code. With SCL decoding, the performance gain  
Figure 6: Bit error rate (BER) performance of a \((8,5)\) polar code with the successive cancellation (SC) and maximum likelihood (ML) decoders. suggests that a longer polar code can provide better error performance in our muli-way relaying system. Furthermore, when comparing SC decoding with SCL decoding, for the same code, we see a significant advantage of the later in the form of coding gains in the range of 1.5-3 dB.