## 5 Conclusions and Future Research Details  
Key research contributions in this paper include: In the context that the existing literature only focuses on the application of PCPE algorithm to even-bit QAM [17] or even-bit QAM based on different probabilistic shaping models [19; 20] signals, this study investigates the compatibility between the PCPE algorithm and traditional cross-shaped odd-bit QAM signals for the first time. Results from qualitative and quantitative analyses confirm that the absence of high-power constellation points at the four corners of the constellation diagram is the primary reason for the decline in recovery accuracy of the PCPE algorithm when applied to odd-order QAM signals. Building upon the aforementioned analysis, this paper introduces, for the first time, a geometric shaping optimization scheme tailored for the PCPE algorithm. Inspired by previous works [22; 24], the proposed scheme achieves optimization by defining the objective function in the PO iteration process as the MI performance index post-recovery of the PCPE algorithm under specific channel conditions. This approach ensures an optimized design that comprehensively considers the visibility of the Post-Squared Constellation's PC vector while ensuring that the MED of the constellation is not excessively small. After the PO iterative convergence, the overall structure of the optimized GS-QAM constellations obtained tends to be closer to a square,  
Figure 15: MI versus SNR for odd-bit QAM constellations with the cascading PCPE-BPS algorithms. (a) 32 QAM and (b) 128 QAM. rather than to a circle like the existing schemes optimized for AWGN only [28; 29; 30], so it can be suitable for the working principle of the PCPE algorithm. The simulation results have also proved that the scheme in this paper can extend the application scope of the PCPE algorithm from only supporting even-bit QAM to full QAM orders. In summary, this paper studies for the first time the compatibility between the PCPE algorithm and odd-bit QAM signals, and points out that the traditional cross-shaped constellation structure will lead to a serious decrease in the estimation accuracy of the PCPE algorithm under high SNR range. Correlation quantitative analysis shows that even for the R-128 QAM signal that is only subjected to a fixed \(\pi/6\) phase rotation, the PCPE algorithm is seriously incompatible with it in the high SNR range, resulting in an MI loss of up to 0.99 bit/symbol. In response to this question, we present a GS approach tailored for odd-bit QAM constellations, demonstrated through rigorous validation via numerical simulation and VPItransmissionMaker simulation. Notably, this study marks the pioneering application of the GS technique to achieve compatibility with the PCPE algorithm. The geometrically shaped signals, when compared to traditional cross-shaped constellation structures, exhibit a substantial enhancement in mutual information performance across the entire measured SNR range. By applying the GS-32 QAM and GS-128 QAM constellation structures designed in this paper to replace the traditional cross-shaped odd-bit QAM signals, the maximum achievable MI enhancement can reach 0.5 bit/symbol and 1.62 bit/symbol, respectively. The proposed constellation design scheme effectively extends the applicability of the low-complexity PCPE algorithm, offering a viable alternative for practical optical communication systems utilizing odd-bit QAM formats. Specifically, due to the clear pursuit of high speed and low power consumption in DCI interconnection scenarios, the proposed scheme can make it possible to reliably apply the low-complexity PCPE algorithm in the odd-bit QAM communication system with high spectral efficiency. Therefore, the proposed scheme will bring significant advantages in this particular communication system. In future investigations, our focus will delve into the following aspects: 1.