## 8 System Implementation Experimental Studies
### System Simulation Results  
The experimental verification of error decoding performance that can be achieved with the proposed QC-RL-LDPC implementation was carried out by simulating a system as in Figure 4. In the results presented in this section, we assume a QPSK modulation (two bits per symbol) and an additive white Gaussian noise (AWGN) channel, with a broad range of simulated signal-to-noise ratios (SNRs). Correction performance is evaluated with Monte Carlo simulations, with the number of messages such that at least 100 of them are in error after HR part decoding, for every SNR.  
In Figure 9, we provide simulation results in the form of goodput performance, which is defined as the ratio of the number of information bits correctly delivered to the total number of QPSK symbols transmitted, required for successful decoding [11]. Obviously, the higher the SNR, the lower the number of average transmitted chunks needed, which means the goodput is higher. The upper limit of goodput for QPSK is \(2R_{\max}\), since if only the initial HR coded vector is transmitted, then \(2R_{\max}\) information bits are successfully delivered per every QPSK symbol. The chunk size is assumed to be equal to \(P\), because the encoding is performed in blocks of \(P\) bits.  
We obtained the parity check matrix \(\mathbf{H}1\) by lifting the protograph taken from [13] (Equations (11) and (12) therein), with submatrix size \(P=16\). Since the minimum achievable rate of this code is relatively high, \(R_{\min}=0.4\), the correction performance of the code \(\mathbf{H}1\) in the low-SNR regime can be improved by proposing codes with more chunks (or rows in the IR part). To address this, we constructed matrix \(\mathbf{H}2\) using the procedure outlined in Section 7, with an increased number of chunks and lower \  
As a point of comparison to other methods, we conducted simulations of a rate-adaptive scheme that utilizes 5G codes. Specifically, we examined BG1 with information block lengths of 132, 264 and 352. It can be observed in Figure 9 that the performance is similar to our designed codes **H**3, **H**2 and **H**5, respectively. This proves that the code construction part of this research provides codes that are not worse than state-of-the-art industrial LDPC codes. However, the advantage of the proprietary QC-RL-LDPC design is that the code parameters of HR code rate, information block length and submatrix size are flexible and can be freely chosen, while the 5G code family offers only two HR code configurations (BG1/BG2) and a discrete set of submatrix size/block length pairs.  
The designed QC-RL-LDPC codes successfully achieve the objective of the rate self-adaptability processing. As shown in Figure 10, the number of chunks required for error-free reception increases as the SNR decreases. The code should be selected for a specific use case, and the algorithm developed can design codes for diversified needs of information block length \(K\), rates \(R_{\text{max}}\), \(R_{\text{min}}\) and chunk sizes \(P\).  
Another decision that needs to be made in the system design is choosing the iteration limits for decoding \(I_{\text{HR}}\) and \(I_{\text{IR}}\) (as defined in Section 6). This selection involves a trade-off between computational burden and decoding performance. However, as illustrated in Figure 11, which shows results for the same set of received input vectors, decoded with different parameters \(I_{\text{HR}}\) and \(I_{\text{IR}}\), the choice of these parameters does not have a very significant impact on the performance. For the system simulation in this paper, we used \(I_{\text{HR}}=20\) and \(I_{\text{IR}}=7\).  
Figure 9: Goodput of the designed codes obtained by Monte Carlo simulation with AWGN channel and QPSK modulation.  
Figure 11: Goodput of the code **H5** with different decoding iteration limits.  
Figure 10: Number of chunks transmitted for different SNR levels for code **H3** in simulation of transmission of 200 messages.