###### Abstract
As described in detail later, for the regular cross-shaped 32 QAM and 128 QAM constellations, due to the lack of signal points at the four corners, the PCPE algorithm will not be able to accurately estimate the phase rotation, resulting in severe achievable information rate (AIR) loss, especially when the SNR is high. In this investigation, we conducted qualitative and quantitative simulations to explore the previously unaddressed issue of compatibility between the PCPE algorithm and the conventional cross-shaped constellation structure. Our findings reveal a noteworthy performance discrepancy, particularly concerning a 128 QAM signal subjected to a slight phase shift of \(\pi/6\), resulting in a substantial MI performance degradation of 0.99 bit/symbol attributed to the convergence error of the PCPE algorithm. Consequently, we introduced a tailored signal constellation design strategy specifically designed for odd-bit QAM constellations, leveraging geometric shaping (GS). Through iterative optimization utilizing the pair-wise optimization (PO) algorithm and setting the objective function as the MI performance of the PCPE algorithm under specific SNR and line width conditions, we effectively addressed the inherent compatibility challenges between the PCPE algorithm and odd-bit QAM formats. To the best of our knowledge, this is the first time that the GS scheme has been studied to adapt the odd-bit QAM signals to the PCPE algorithm. Both numerical simulation and VPItransmissionMaker results consistently demonstrate that the proposed GS constellations are notably more suited for the PCPE algorithm, yielding superior mutual information performance across the entire measured signal-to-noise ratio (SNR) range. The simulation results show that the GS-32 QAM and GS-128 QAM constellation structures proposed in this paper can achieve MI performance gains of up to 0.50 bit/symbol and 1.62 bit/symbol, respectively, compared with the traditional cross-shaped constellation structure. Furthermore, the utilization of the proposed GS constellations and the PCPE algorithm substantially reduces the required SNR to achieve the 20% soft-decision forward error correction (SD-FEC) threshold, when compared to conventional cross-shaped constellations and either BPS or PCPE algorithms. Specifically, the proposed GS-32 QAM scheme can achieve SNR gains of 1.10 dB and 1.60 dB compared to the schemes using the BPS algorithm and the PCPE algorithm for regular 32 QAM, respectively. The remainder of this paper is organized as follows. Section 2 provides the principles of the PCPE algorithm, qualitatively and quantitatively analyzes the interplay of it and odd-bit QAM signals, and introduces a pairwise optimization (PO) algorithm-based GS QAM generation procedure, which can design odd-bit QAM constellation structures suitable for the PCPE algorithm. In Section 3, the MI performance of the proposed scheme is verified based on both Monte Carlo numerical simulation and commercial simulation software. Section 4 is a detailed discussion of the innovations and results of this paper, including the design rationale of the proposed scheme, sensitivity analysis in different channel conditions, and compatibility with a cascading algorithm named PCPE-BPS. Section 5 summarizes our work and provides an outlook for future research details.