## 5 Conclusions  
This paper considers polar codes with DBPSK when used as network codes in multi-way relaying. The use of DBPSK bypasses the need for channel estimation at the receiver side in such systems. This system can be viewed as a systematic linear block code. We present a methodology of using equivalent systematic polar encoding in this system, as well as the the SC and SCL decoding algorithms. Furthermore, this paper also considers the encoding and decoding complexities of the system for linear block codes with ML decoding, LDPC codes with log-BP decoding, and polar codes with SC and SCL decoding. It is shown that polar codes with SC or SCL decoding have a much lower complexity than LDPC codes with log-BP decoding, making them attractive for practical multi-way relaying. We use computer simulations to assess the error rate performance of multi-way relaying employing polar codes with different code lengths and rates. Monte-Carlo simulation results indicate that the system employing polar codes has a significant performance gain compared to an uncoded scheme. For short polar codes, SC and SCL decoding can achieve an error performance that is close to ML decoding. For longer polar codes, the error performance is comparable to LDPC codes with log-BP decoding. For practical applications, we consider a hard threshold technique at the terminals to determine whether a relay transmits. This technique can efficiently reduce the performance loss with respect to the ideal case when a terminal knows when a relay transmits with SC and SCL decoding. When compared with the use of LDPC codes in a multi-way relaying system employing log-BP decoding, polar codes provide similar performance gains. In some cases when the LDPC codes have relevant groups of terminals of small sizes, and hence are more suitable for multi-way relaying, polar codes show a small performance loss of about 1.7 dB in practical cases. Notice that the polar codes we considered in this paper are standard and not specifically designed for multi-way relaying. It is expected that polar codes that are specifically designed for multi-way relaying could provide further performance gains; hence, this topic is of considerable interest for further research. In particular, multi-kernel polar codes offer extra flexibility that could be very appealing for multi-way relaying applications. Even with the standard binary polar codes we used in this work, the advantages of low-complexity  
Figure 11: Bit error rate (BER) comparison of \(\hat{\mathbf{m}}_{T}\) and \(\hat{\mathbf{m}}_{A}\) of a \((512,416)\) polar code with successive cancellation list (SCL) decoding.  
decoding offset the small losses and make polar codes a practical coding technique for multi-way relaying, which is competitive to LDPC codes. While we consider polar codes in this paper, the technique of using a systematic equivalent code for transmission and then a corresponding non-systematic code with structural features that allows for simplified decoding for information retrieval can be used with any linear block code. This method increases the class size of linear block codes that can be used as network codes in multi-way relaying to also include non-systematic codes.  
Conceptualization H.L.; methodology H.L.; software R.J.; investigation R.J. and H.L.; writing--first draft R.J. and H.L.; writing--review and editing, H.L.; supervision, H.L.; funding H.L. All authors have read and agreed to the published version of the manuscript.  
This research was supported by grant RGPIN-2016-03647 awarded by the Natural Sciences and Engineering Research Council of Canada (NSERC).  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.