## 3 Results  
### Monte Carlo Simulation Results  
To further evaluate the performance of the proposed geometric shaping constellation designs under different SNRs, and compare it with the performance of regular constellations, a numerical simulation was carried out. In the simulation system, the symbol rate of the odd-bit QAM signal was 32 GBaud, while the total linewidth of the lasers on the transmitting end and receiving end combined was 200 kHz. Cycle slips were ideally compensated to simplify the MI computation, as [17]. Each MI value was an average obtained through 32 realizations with 1024 blocks of 64 symbols.  
Figure 6 shows the MI performance using the GS constellations and PCPE algorithm, as well as the MI performance using the regular constellations and two different phase recovery algorithms, namely the BPS algorithm and the PCPE algorithm. For the BPS algorithm, a two-stage implementation is adopted, and the number of test phases in each stage is set to 11, which realizes the effect of searching among a total of 121 test phases. This implementation and parameter setting are consistent with [17].  
As shown in Figure 6a, for a regular cross-shaped 32 QAM constellation, applying the PCPE algorithm can achieve better MI performance than the BPS algorithm in the low SNR value range. However, when the SNR is high, due to the incompatibility between the cross-shaped constellation and the PCPE algorithm, the severe residual phase error will result in a maximum AIR loss of about 0.38 bit/symbol compared with the BPS algorithm. By using the proposed GS-32 QAM constellation, the MI performance advantage of the PCPE algorithm relative to BPS in the low SNR stage can be further expanded, and the AIR loss in the high SNR stage compared to BPS can be effectively reduced to below 0.06 bit/symbol. Throughout the entire evaluated SNR range, by replacing the regular 32 QAM (R-32 QAM) with the proposed GS-32 QAM, the MI performance improvement of the PCPE algorithm  
Figure 5: Constellation diagrams of (**a**) GS-32 QAM before squaring; (**b**) GS-128 QAM before squaring; (**c**) GS-32 QAM after squaring; and (**d**) GS-128 QAM after squaring.  
varies from 3.48% and 15.73%. When considering the threshold corresponding to 20% SD-FEC overhead, the proposed 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.  
Similarly, the MI curves using 128 QAM format are shown in Figure 6b. For the cross-shaped constellation, the AIR loss of using the PCPE algorithm under high SNR compared to using the BPS algorithm is up to 1.30 bit/symbol. After employing the proposed GS-128 QAM constellation, the AIR loss can be greatly reduced to 0.05 bit/symbol. Throughout the entire evaluated SNR range, by replacing the regular 128 QAM (R-128 QAM) with the proposed GS-128 QAM, the MI performance of the PCPE algorithm is improved by a percentage above 7.34% and below 22.17%. When considering the threshold corresponding to 20% SD-FEC overhead, the proposed scheme can achieve an SNR gain of 2.59 dB compared to the traditional scheme using the BPS algorithm and a regular 32 QAM constellation.