## 3 The Generalized AP-HSIC Algorithm for QAM Modulations
Lastly, we included a BER calculator to measure the practical effects of the system's performance. The second simulation also used the MGSTC transmitter and MIMO channel matrix block with the same components as described earlier. However, on the receiver side, instead of an MGSTC decoding algorithm block, we replaced it with an AP-HSIC block to strengthen the receiver's ability to operate in parallel decoding communication mode. The third simulation was based on the same components described in the first and second simulations, but we changed the decoding algorithm from AP-HSIC to LM-HSIC. We examined the systems' architectures regarding the BER vs. average SNR by running the interference Rayleigh deep-fading scenario with high-level AWGN under three modulation orders as follows: 16-, 32-, and 64-QAM at the transmitter. Eventually, we compared the performances between MGSTC and AP-HSIC and between LM-HSIC and AP-HSIC. The simulation performance parameters are determined by [60]. As it is widely recognized, the systematic BER of the MGSTC decoding algorithm always corresponds to the BER of its final iteration. The systematic BER of the MGSTC decoding algorithm was compared to the average total BER of the AP-HSIC and the average total BER of the LM-HSIC, as shown in the following figures. ### MGSTC and AP-HSIC Simulations under Rayleigh Deep-Fading Scenario and High-Leveled AWGN

As described earlier, this section presents simulation results of the algorithms MGSTC and AP-HSIC in the case of deep-fading with high-leveled AWGN based on 16-QAM, 32-QAM, and 64-QAM modulations. In Figure 3, we present graphs showing the BER performance as an average SNR function using 16-QAM constellation modulation under the Rayleigh deep-fading scenario and high-leveled AWGN. These graphs compare the BER performance improvements of the AP-HSIC algorithm with the MGSTC algorithm. They demonstrate the BER performance of the three iterations of MGSTC alongside the overall BER performance of AP-HSIC. Similarly, the graphs at Figure 4a,b illustrate the graphs showing the BER performance of the MGSTC and the AP-HSIC decoding algorithms as an average SNR function under 32-QAM and 64-QAM constellation modulation, respectively, under the Rayleigh deep-fading scenario and high-leveled AWGN. In Figures 3 and 4a,b, we observe that only the first iteration of the MGSTC decoding algorithm (only the \(2\times 2\) MIMO case) achieves a BER performance that enables decoding and extracting information. In contrast, the second and third iterations do not converge and exhibit high BER levels even at an extreme SNR. It can be seen that in Figure 3, the blue graph that represents the BER performance of the first iteration of the MGSTC scheme achieves \(BER\approx 10^{-3}\) at an average \(SNR\approx 16\) dB. The purple graph represents the BER performance of the AP-HSIC based on the full MIMO case \(6\times 6\) and achieves \(BER\approx 10^{-3}\) at an average \(SNR\approx 14\) dB. The yellow graph (which represents the BER performance of the second iteration of the MGSTC system based on the MIMO case \(4\times 6\)) and the orange graph (which represents the total BER performance of the MGSTC system based on the full MIMO case \(6\times 6\)) do not converge to BER levels that enable the reproduction of information. In Figure 4a, the blue graph representing the BER performance of the first iteration of the MGSTC decoding algorithm achieves \(BER\approx 10^{-3}\) at an average \(SNR\approx 20\) dB and achieves \(BER\approx 5\cdot 10^{-4}\) only at an average \(SNR\approx 25\) dB. The purple graph represents the BER performance of the AP-HSIC, achieves \(BER\approx 10^{-3}\) at an average \(SNR\approx 19\) dB, and achieves \(BER\approx 10^{-6}\) at an average \(SNR\approx 25\) dB. The yellow and orange graphs, representing the BER performance of the second and third iterations of the MGSTC, respectively, also do not converge to BER levels that enable the reproduction of information in the 32-QAM modulation.