## 3 The Generalized AP-HSIC Algorithm for QAM Modulations
In Figure 4b, we can see that the blue graph represents the BER performance of the first iteration of the MGSTC scheme an, achieves \(BER\approx 10^{-3}\) at an average \(SNR\approx 24\) dB, and achieves \(BER\approx 10^{-5}\) only at an average \(SNR\approx 30\) dB in the 64-QAM modulation. The purple graph represents the BER performance of the AP-HSIC, achieves \(BER\approx 10^{-3}\) at an average \(SNR\approx 20\) dB, and achieves \(BER\approx 10^{-6}\) at an average \(SNR\approx 28\) dB in the 64-QAM modulation. The BER performance of the second and third iterations of the MGSTC, represented by the yellow and orange graphs, also fails to converge to BER levels that enable the reproduction of information in the 64-QAM modulation. In our series of experiments, we have gained important insights and drawn conclusions. When comparing the BER performance of the AP-HSIC algorithm with the second and third iterations of the MGSTC decoding algorithm under scenarios of Rayleigh deep-fading in three types of modulation order, it is evident that the AP-HSIC algorithm outperforms the others. It exhibits superior BER performance and an efficient parallel decoding process. This is due to the fact that the second and third iterations of the MGSTC decoding algorithm fail to achieve the required BER performance for decoding the transmitted information (see Figure 5). The proposed AP-HSIC offers better diversity gain compared to the first iteration of the MGSTC algorithm, despite the fact that the transmission power of the first two antennas in the first iteration of the MGSTC is six times higher than in the full MIMO mode of the AP-HSIC scheme (\(6\times 6\) in our proposal compared to \(2\times 2\) in the first iteration of the MGSTC scheme). In Figure 3, it can be observed that AP-HSIC achieves a diversity gain \(\approx 2\) dB higher compared to the first iteration of the MGSTC. In Figure 4a, the AP-HSIC presents a diversity gain \(\approx 2\) dB higher compared to the first iteration of the MGSTC. In Figure 4b, the AP-HSIC presents a diversity gain \(\approx 5\) dB higher compared to the first iteration of the MGSTC. In the proposed scheme, the BER performance is significantly improved compared to the first iteration of the MGSTC scheme. There is also an enhancement in spatial utilization, and, equally important, the energy efficiency of the AP-HSIC is significantly higher than that of MGSTC. Additionally, the ability of the AP-HSIC to mitigate interference when combined with parallel decoding (i.e., it allows parallel decoding and utilizes the operation of all six transmission/receiving antennas in parallel) is superior to the serial decoding and error accumulation of the MGSTC. The AP-HSIC scheme consistently shows lower BER levels at lower average SNR areas for all constellation orders than the MGSTC scheme. Furthermore, it is worth noting that MGSTC has a significant limitation: errors accumulate with each iteration. This means that the accuracy of each iteration sets a lower bound for the accuracy of subsequent iterations. On the other hand, AP-HSIC, which relies on a parallel decoding process, consistently maintains robust BER performance across all decoding processes. In Figure 5, we have conducted a detailed comparison of the systematic BER performance of MGSTC and AP-HSIC algorithms across different average SNR levels. We tested all three constellation orders under the Rayleigh deep-fading scenario and high-level AWGN. The systematic BER performance of the MGSTC algorithm does not converge to a suitable level for minimal normal decoding across all three scenarios. Conversely, the BER performance of the AP-HSIC system consistently converges to a level suitable for decoding in all three scenarios. Figure 6 presents the number of iterations in the main loop of the AP-HSIC algorithm for each SNR value in 16-QAM modulation under the Rayleigh deep-fading scenario and high-level AWGN. Clearly, higher SNR values result in faster convergence.