## 1 Introduction
In many cases, the Rayleigh deep-fading phenomenon significantly decreases the BER performance and increases the number of decoding iterations of the algorithms. All the algorithms we have reviewed so far in this article do not handle the Rayleigh deep-fading scenarios. Another critical aspect that advanced decoding algorithms such as LCMPD, MPD, and LLR must focus on and treat is the exact identification of the statistical distribution of the interference and the noise parts. These algorithms generally establish a basic assumption that the noise components received in the receiver and the matrix of transmitted symbols have a Gaussian distribution [1; 29]. This reference makes it possible to calculate the interference component's duration and variability and estimate the conditional probability function of editing a transmitted symbol matrix given the channel matrix and the received signal matrix in the decoding process. Therefore, if there is an interference with a non-Gaussian distribution or a channel medium that causes the distribution of the transmitted signal to change [15; 45], it will create many errors in the calculation processes of these algorithms. This paper proposes a generalized version of the new spatial SIC decoding algorithms--the AP-HSIC [56]--that only requires the CSIR assumption and generates computational feedback at the receiver. This computational feedback can overcome the effects of random scatters in a multi-path fading channel and several interference effects, such as Rayleigh deep-fading under quasi-static flat Rayleigh fading MIMO channels combined with high-leveled AWGN. In this paper, we present the algorithm's BER performance of three systems based on three spatial SIC decoding algorithms and compare them. The first system, the MGSTC spatial SIC decoding algorithm, is based on the MIMO array's serial-decoding mode. The second system is based on the AP-HSIC algorithm applied to \(M\)-QAM, which combines parallel processing decoding methods in the MIMO array. The last system is the proposed LM-HSIC algorithm, based on Lagrange multipliers optimization and parallel decoding. This comparison was conducted in an environment of different Rayleigh deep-fading, high-level AWGN, and scatter scenarios. The comparison is reflected in the performance levels of BER vs. the average Signal-to-Noise Ratio (SNR). The analysis considers different high constellation orders (16, 32, and 64-QAM) in the three wireless system networks (MGSTC, AP-HSIC, LM-HSIC). We also present the theoretical graphs of the BER vs. SNR of the three constellation orders with AWGN only and the BER vs. SNR graphs of the three constellation orders with fading without an error correction mechanism at the receiver. We compared all these graphs with the BER vs. SNR graphs of the three constellation orders with the AP-HSIC performance under fading scenarios. The proposed AP-HSIC and LM-HSIC algorithms for M-QAM offer an advanced solution for effectively canceling digital interference. This innovative approach provides unprecedented flexibility for SA networks without the need to control the channel or share a channel that statistically measures the spatial domain. In addition, the proposed algorithms enable the real-time decoding of symbol matrices in a parallel MIMO mode, ensuring seamless interference cancellation without compromising the decoding rate. These capabilities are a game-changer for ultra-high bit-rate and high-capacity channel requirements. Withthe proposed algorithms, we can differentiate between the original channel-response matrix and interfering factors, \(\Delta H\), allowing for the accurate calculation and timely updating of the general MIMO channel-response matrix online. The computation of \(\Delta H\) allows a total offset of the interference/deep-fading effects. As a result, the system can decode the original symbols without additional transmission power consumption or the need for re-transmission.