# Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping
## 5 Performance Comparison
### End-to-End Decoding Performance  
In the following, the decoding performance of PAS is evaluated after transmission of 64-QAM over the AWGN channel. The BRGC in Figure 2 (bottom) is used for amplitude to bit mapping after shaping, and for symbol mapping after FEC encoding as shown in Figure 2 (top). Different transmission rates, and codeword length regimes of LDPC codes are considered. For each SNR, the simulations are run until at least 100 frame errors are observed. For the first case of long FEC codes, we use the LDPC codes from the DVB-S2 standard [77] with codeword length \(n_{\text{c}}=64800\) bits. In the case of short FEC codes, the LDPC codes from the IEEE 802.11 standard [78] with codeword length \(n_{\text{c}}=648\) bits are used. Maximum 50 iterations are performed during the belief propagation decoding at the receiver. For a fixed 1-D constellation size \(M=2^{m}\), FEC code rate \(R_{\text{c}}\) and target transmission rate \(R\), we compute \(\gamma=R_{\text{c}}m-(m-1)\) and accordingly, \(k/n=R-\gamma\). Here, the total number of 1-D symbols in an \(n_{\text{c}}\)-bit FEC codeword is \(n\). For DM algorithms working with \(\mathcal{A}=\{1,3,5,7\}\), the AWGN-optimal MB PMFs at 10.7 and 14 dB SNR are quantized to obtain the integer composition based on [93] for the  
Figure 9: Finite-length information rate vs. SNR for various shaping architectures at \(n=216\). target rates 2 and 2.25 bit/1-D, respectively. For SpSh algorithms, \(E^{\bullet}\) is selected as the minimum value that satisfies \(R_{\rm s}\geq k/n\). Both ESS and SM are then implemented with full-precision. Figure 10 shows the decoding performance with the LDPC codes from DVB-S2 for ESS, SM, MPDM, and uniform signaling at a transmission rate of 4.5 bits per complex channel use (bit/2-D). ESS, SM and MPDM, all of length \(n=180\) amplitudes, use either the LDPC code of rate \(R_{\rm c}=5/6\) (solid curves) or rate \(R_{\rm c}=4/5\) (dashed lines). In order to achieve a transmission rate of 4.5 bit/2-D, the redundancy added by the shaping scheme is varied. For rate-5/6-coded signaling, \(\gamma=0.5\) and thus, \(k/n=1.75\) bit/1-D. For rate-4/5-coded signaling, \(\gamma=0.4\) and thus, \(k/n=1.85\) bit/1-D. For uniform 64-QAM, the FEC code rate is set to \(R_{\rm c}=3/4\). We observe for shaped schemes that the performance with FEC code of rate 5/6 is superior to rate 4/5 as predicted in Figure 5, for which the reasons are outlined in Section 3.4. Therefore, we will focus on \(R_{\rm c}=5/6\) in the following. In Figure 10, at a FER of \(10^{-3}\), SpSh and MPDM outperform uniform signaling by approximately 0.9 dB. We further note that ESS, SM and MPDM have very similar performance, with ESS and SM being approximately 0.05 dB more power-efficient than MPDM. This is in good agreement with the rate loss analysis of Figure 8 where also only a marginal improvement of the SpSh schemes over MPDM is found. Finally, as discussed in Section 5.1, there is no visible difference in performance for ESS and SM at \(n=180\). For short LDPC codes, the shaping blocklength is set to \(n=216\). In Figure 11, the decoding performance is shown at transmission rates of 4 and 4.5 bit/2-D. Uniform 64-QAM is encoded with LDPC codes of rate 2/3 and 3/4, respectively. For shaped signaling, the code rates that minimize \(\Delta\)SNR for 64-QAM at rates 4 and 4.5 bit/2-D are computed to be \(R_{\rm c}\approx 0.79\) and 0.83 using (13), respectively. Therefore, being the closest available to these values, \(R_{\rm c}=5/6\) is used for shaped signaling. Figure 10: FER vs. SNR for 64-QAM at a transmission rate of 4.5 bit/2-D. Rate-\(R_{\rm c}\) DVB-S2 LDPC codes of length \(n_{\rm c}=64{,}800\) bits are used. All shaping schemes use a blocklength of \(n=180\). At this shaping blocklength, each LDPC codeword consists of 120 shaped blocks. As shown in Figure 11, at rates 4 and 4.5 bit/2-D, SpSh and MPDM perform similarly, and require 1.1 and 0.9 dB less SNR than uniform signaling to achieve a FER of \(10^{-3}\), respectively.