# Soft Decision Decoding with Cyclic Information Set and the Decoder Architecture for Cyclic Codes
## 6 Conclusions  
A soft decision decoding algorithm based on a cyclic information set is proposed, and its efficient hardware implementation architecture is also presented. The algorithm employs the correlation discrepancy as the decoding metric and adopts an iterative computation method. Firstly, the cyclically generated equivalent information sequences, which have the minimum correlation discrepancy with the received sequence of channel observation values, are selected as the candidate information sequences. Then, the list of candidate codewords is encoded using the very concise encoder. Finally, the optimal decoding result is obtained by selecting the optimal candidate codeword using the correlation discrepancy metric. The method reduces the search range of the decoding results by establishing the list of candidate codewords in the cyclic and iterative manner, and the complexity is lower than the ML algorithm. Furthermore, in order to verify the performance of the decoding  
Figure 8: The BER performance of (31, 21) BCH code when SNR is 5 dB. **(a)** The BER performance with different \(I\) and \(C\). **(b)** The BER performance with different candidate codeword list sizes, \(I\cdot C\).  
Figure 7: The BER performance of (23, 12) Golay code when SNR is 5 dB. **(a)** The BER performance with different \(I\) and \(C\). **(b)** The BER performance with different candidate codeword list sizes, \(I\cdot C\).  
Figure 9: The BER performance of (63, 30) BCH code when SNR is 4 dB. **(a)** The BER performance with different \(I\) and \(C\). **(b)** The BER performance with different candidate codeword list sizes, \(I\cdot C\).  
algorithm, the implementation architecture of the algorithm is completed on the FPGA platform. The BER performances of the (15, 5) BCH code, (31, 21) BCH code, (63, 30) BCH code, and (23, 12) Golay code are evaluated on this platform. The experimental results show that the performance of the proposed decoding algorithm is close to the ML algorithm, and the decoder achieves a good trade-off between decoding complexity and error correction performance. The effect of the number of information sequences and the candidate codewords on the decoding performance is also evaluated on the FPGA platform. The complexity of the decoder can be reduced by optimizing the number of information sequences and candidate codewords.  
In this paper, we only evaluated the BER performance of the cyclic codes over the AWGN channel. In the future, considering short cyclic codes have been widely employed in 5G communications, it is valuable to evaluate the proposed decoding algorithm under the wireless fading channel, combined with advanced modulations or waveforms. In addition, the serial hardware implementation architecture has been utilized to verify the BER performance of the proposed algorithm. In the future, efficient parallel architectures will be developed to meet the higher throughput requirements of 5G, enabling practical applications.  
Conceptualization, W.C. and C.H.; methodology, W.C. and C.H.; software, W.C. and T.Z.; validation, W.C. and C.H.; formal analysis, W.C. and T.Z.; investigation, W.C. and C.H.; writing--original draft preparation, W.C. and T.Z.; writing--review and editing, W.C., T.Z. and C.H. All authors have read and agreed to the published version of the manuscript.  
This research is partly supported by the Seed Fund of Tianjin University (No. 0903061008).  
Data Availability Statement: Data are available upon request.  
The authors thank the former masters student [PERSON] in [PERSON]'s group, who wrote part of the programs and prepared some raw data. We also thank the former masters student [PERSON] in [PERSON]'s group, who prepare part of the figures and the data analysis. All authors have read and agreed to the published version of the manuscript.  
Author [PERSON] has been authorized for the related methods in two Chinese patents (CN201510846874.3, CN201611099566.X).