Amplitude Distribution
By Constant Composition Distribution Matching (CCDM), $k_\text{c}$ uniformly distributed data bits are transformed into $n_\text{c}$ shaped amplitudes: $$D^{k_\text{c}}\text{---}\boxed{\text{CCDM}}\text{---}A^{n_\text{c}}$$
• Data bits $D_i$ iid Bernoulli(1/2).
• The $A_i$ have MB distribution on amplitude set $\{1,3,\dotsc,2^m-1\}$: $$P_A(a)=\frac{e^{-\nu a^2}}{\sum_{\tilde{a}\in\mathcal{A}}e^{-\nu \tilde{a}^2}}$$
• For large enough $n_\text{c}$, the CCDM rate is equal to the entropy of $A$, i.e., $k_\text{c}/n_\text{c}=H(A)$.
• CCDM is invertible: receiver recovers $D^{k_\text{c}}$ from $A^{n_\text{c}}$ with zero error.

P. Schulte and G. Böcherer, Constant composition distribution matching, IEEE Trans. Inf. Theory, vol. 62, no. 1, pp. 430–434, Jan. 2016.