![image](parity_constellation_combined_subfigure)

Probabilistic Eigenvalue Shaping {#sec:shaping_time_share}
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In the previous section, we observed a significant gap between the time-scaled of the system in [@shevchenko2018tcom] and the capacity. This gap is referred to as shaping gap. In order to close it, we propose a system as shown in [Fig.]{}, inspired by  [@bocherer2015_bw_efficient_rate_matched_ldpc].

In the scheme, the sequence of uniformly distributed data bits is mapped to a sequence of positive amplitudes distributed half Gaussian by a . The binary image of this sequence is encoded by a systematic code, resulting in uniformly distributed parity bits, which are then used to map the sequence of half Gaussian distributed symbols to a stream of Gaussian distributed symbols.

As the capacity-achieving distribution $p_X^*(\lambda)$ is not symmetric, cannot be directly applied here. However, in order to keep the benefits of , we wish to apply the before the . We describe in the following with reference to [Fig.]{}. The binary data sequence $\bm{u}$ of length $k_\mathsf{s}$ bits is mapped by the to a sequence of eigenvalues $\bm{\lambda}\in\Lambda^{n_\mathsf{s}}$ of length $n_\mathsf{s}$ distributed according to $p_X^*(\lambda)$. The can be used for that purpose [@schulte2016_ccdm]. It is asymptotically optimal as its rate $R_\mathsf{s}$ approaches the entropy of the desired channel 