## 3 The Generalized AP-HSIC Algorithm for QAM Modulations
This computational feedback is essential for advancing MIMO technology in modern wireless communications, especially for technologies relying on Ultra-Reliable Low-Latency Communication Coding (URLLC) [61], some of which must be able to respond instantaneously, such as robots, autonomous vehicles, and medical equipment. Moreover, some technologies physically do not allow feedback (e.g., satellite communications) [61]. Therefore, computational feedback is critical when applying these technologies. The simulation results showed that there is a necessary balance between energy efficiency and computational capabilities. The findings suggest that merely boosting transmission power does not always decrease the BER when faced with different interference scenarios, especially when there is a requirement to enhance the amount of information and transmit at a higher data rate (such as when higher modulation is necessary). Equally important, the ability to decode a heavy information transmission in parallel spatial decoding mode and simultaneously to SIC in complex interference scenarios is critical for the next generation of modern wireless network MIMO systems. The diversity of SIC decoding algorithms, such as AP-HSIC and LM-HSIC, enables real-time decoding capabilities in the real world of advanced MIMO techniques and allows them to break through the limitations of real-time space-SIC algorithms. In our study, we observed a significant improvement in decoding performance by analyzing the BER versus SNR graphs of AP-HSIC and LM-HSIC algorithms. We compared these algorithms to the decoding algorithm MGSTC when dealing with destructive interference phenomena such as Rayleigh deep-fading and high-level AWGN. Our theoretical analyses and simulations enabled us to highlight the main contributions of the AP-HSIC and LM-HSIC decoding algorithms, which are as follows: Improved BER versus SNR performance, resulting in significantly enhanced diversity gain and diversity order of the

\[=\left\|Y\big{(}I-S^{+}S\big{)}\right\|_{F}^{2}+\left\|\big{(}YS^ {+}-(H+\Delta H)\big{)}S\right\|_{F}^{2},\]8 Comparisons between BER performances vs. the average SNR of AP-HSIC under the scenarios of Rayleigh fading–SISO with 16-, 32-, and 64-QAM and the theoretical 16-, 32-, and 64-QAM SISO AWGN only, and between the scenarios of Rayleigh fading SISO 16-, 32-, and 64-QAM without any correction algorithm. AP-HSIC and the LM-HSIC compared to MGSTC, especially in relation to the second and third iterations of MGSTC, which fail to converge to a minimum BER for information decoding. The AP-HSIC and LM-HSIC algorithms play a crucial role in enhancing energy efficiency. A more efficient communication system combining the AP-HSIC and LM-HSIC decoding algorithms, allowing for the transmission of six times less transmission energy per symbol compared to the first iteration of MGSTC (which has an order antenna array of \(2\times 6\)), three times less transmission energy per symbol compared to the second iteration (order antenna array of \(4\times 6\)), and \(1.5\) times less transmission energy per symbol compared to the third iteration (order antenna array of \(6\times 6\)). This is achieved while maintaining a fixed-order antenna array of \(6\times 6\) antennas in the AP-HSIC and LM-HSIC without needing orthonormal matrix bases to mitigate interferences, which are ineffective in Rayleigh deep-fading scenarios. Additional contributions are the ability of AP-HSIC and LM-HSIC algorithms to decode information in parallel compared to the serial decoding of MGSTC, as well as the capability of computational feedback without the need for closed-loop MIMO capabilities, enabling the handling of diverse interference scenarios such as Rayleigh deep-fading. In our upcoming research, we aim to enhance the concept of computational feedback, examine assumptions regarding selective channels in frequency and time-space, and explore non-stationary channels in a comprehensive manner.