# Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping
## 7 Conclusions  
This paper reviewed prominent amplitude shaping architectures and algorithms for the probabilistic amplitude shaping (PAS) framework. Constant composition distribution matching (CCDM), multiset-partition DM (MPDM) and sphere shaping (SpSh) are all optimum shaping techniques for asymptotically large blocklengths, in the sense that they have vanishing rate loss. However, for short blocklengths, CCDM addresses a smaller set of output sequences than that of MPDM and SpSh, leading to higher rate losses. We provided evidence for the AWGN channel that seeking to utilize the signal space in energy-efficient manners is better than attempting to obtain the capacity-achieving distribution, which is derived for asymptotically large, and thus, impractical blocklengths. Therefore, MPDM, SpSh, and other energy-efficient shaping architectures are suitable to be used over a wider blocklength regime, especially for blocklengths below a couple of hundred symbols. In addition to the rate loss analysis, we evaluated information rates and frame error rates (FER) of PAS employing CCDM, MPDM and SpSh as the amplitude shaping architecture. Enumerative sphere shaping (ESS) and shell mapping (SM) are both considered as potential SpSh algorithms. AWGN channel simulations with 64-QAM demonstrate that power-efficiency gains on the order of 1 dB can be obtained already at blocklengths around 200 by employing MPDM and SpSh, and thus, justify our earlier observation on the objective of amplitude shaping. CCDM provides gains around 0.75 dB for the same settings. Furthermore, these gains are predicted well by shaping gain and information rate computations based on bit-metric decoding. Figure 12: Approximate illustration of maximum computational complexity vs. maximum required storage for ESS, SM and CCDM. Red- and blue-colored markers indicate FP and BP implementations, respectively. Radii of the markers are proportional to the corresponding blocklength \(n\in\{64,216,512\}\). Here we assume that BP AC-CCDM is implemented with finite-precision arithmetic using 16-bit numbers which is comparable to the values selected in [72]. Furthermore, we assume that a \(k\)-bit addition is equivalent to \(k\) bit operations, while a \(k\)-bit multiplication is equivalent to \(k^{2}\) bit operations as in [23]. In the last part of the paper, we discussed the performance of shaping algorithms considering latency, required storage and computational complexity. To realize DM, arithmetic coding (AC)-based implementation of MPDM requires minimal storage and can be implemented with a few computations per input symbol. However, AC has a higher serialism than subset ranking (SR)-based implementation which on the other hand has increased computational complexity. For SpSh, ESS and SM provide a tradeoff between storage and computational complexities, where the complexity is more due to the required storage for ESS and required number of computations for SM. Thus, the decision on which algorithm should be used to realize energy-efficient amplitude shaping depends on the application-specific requirements on latency, available storage and tolerable computational complexity. Conceptualization, Y.C.G., T.F., A.A., and F.M.J.W. ; Investigation, Y.C.G., T.F., A.A., and F.M.J.W. ; Writing--original draft, Y.C.G., and T.F. ; Writing--review & editing, Y.C.G., T.F., A.A., and F.M.J.W. All authors have read and agreed to the published version of the manuscript. The work of [PERSON] and [PERSON] has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 757791). The authors would like to thank [PERSON] (Eindhoven University of Technology) for his insightful comments on the complexity discussion in Section 6, and to thank [PERSON] (Sant'Anna School of Advanced Studies, Pisa, Italy) for fruitful discussions. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: