## 4 Hardware Implementation and Result Discussion
### Analysis of the Decoding Performance  
In order to directly reflect the effectiveness of the priority-based schedule and double update queues in improving decoding performance, we made a Monte Carlo simulation to obtain the frame error rate (FER) curves in the AWGN channel as shown in Figure 6. One million codewords are sent for each SNR. The maximum iteration number was set to 10 and the modulation format was set to quad-phase shift keyed (QPSK). The PCM is chosen from 5G NR with code rate 22/27. The simulated priority-based decoder with a single update queue, priority-based decoder with double update queues, residue-based decoder with a single update queue, and residue-based decoder with double update queues are all with 13 pipeline stages. All these four decoders were implemented as an offset min-sum decoder with LLRs quantized to eight bits and messages quantized to six bits, as [15] did. In order to compare the decoding performance fairly and reflect the decoding performance accurately, we take FER = \(10^{-5}\) as the standard as [15] did. From Figure 6, it is apparent to see that double update queues significantly improve the decoding performance. The priority-based decoder with double update queues shows a gain of 0.4 dB compared with the one with a single update queue. The residue-based decoder with double update queues achieves a gain of 6.5 dB compared with the one with a single update queue, since some LLRs of variable nodes can never be updated with a single update queue during decoding. From Figure 6, it can also be found that the priority-based decoder needs lower  
Figure 5: Percentage of updated LLRs per iteration during decoding as a function of number of pipeline stages for rate 22/27 from 5G NR. HS: Decoder with the hybrid schedule. SNR than the residue-based decoder when achieving the same decoding performance because it updates LLRs more frequently. Another Monte Carlo simulation was also done in the AWGN channel for 5G NR (code rate 22/27) and WiMAX (code rate 3/4) to show the results between the SNR and frame error rate (FER) of various decoders, as shown in Figure 7. In the Monte Carlo simulation, one million codewords are sent out to calculate the FER for each SNR. The simulation for 5G NR was performed for different maximum iteration numbers (\(it_{max}=10\), \(it_{max}=20\) and \(it_{max}=30\)). The simulation for WiMAX was performed when the maximum iteration number was set to 10. For a fair comparison with [15], the number of pipeline stages in 5G NR was set to 13, as [15] did. The number of pipeline stages in WiMAX was set to 10. The detail of the hardware implementation will be discussed in Section 4.3. The algorithm and quantization of LLRs and messages were also set as [15] did, where the algorithm was OMSA, the LLRs are quantized as eight bits and messages are quantized as six bits. The modulation format was set to QPSK. For a fair comparison with the hybrid decoding, we take FER = \(10^{-5}\) as the standard as [15] did. In Figure 7a, it can obviously be seen that for 5G NR the loss of SNR performance between the layered decoder and the priority-based decoder with double update queues is 0.2 dB. The loss of SNR performance between the layered decoder and the hybrid decoder is 0.6 dB. Thus, the decoding performance loss at FER = \(10^{-5}\) narrowed from 0.6 dB to 0.2 dB by using the priority-based layered schedule with double update queues when the maximum iteration is set to 10. When the maximum iteration is set to 20 and 30, the loss of SNR performance does not exist. In order to reflect the improvement of the double update queues, performance of residue-based layered decoder with double update queues is also simulated. The loss of the residue-based decoder is just 0.3 dB after using the double update queues at 10 iterations. Note that the priority-based layered schedule has a faster convergence than the residue-based layered schedule.