input $X$, $$\begin{aligned}
    R_\mathsf{s} &=\frac{k_\mathsf{s}}{n_\mathsf{s}} \rightarrow {\mathbb{H}(X)} \text{  as  } n_\mathsf{s} \rightarrow\infty.\end{aligned}$$ For large block sizes, the gap between $R_\mathsf{s}$ and ${\mathbb{H}(X)}$ is sufficiently small and can be neglected. Note that some of the possible eigenvalues may occur with probability zero.

We consider the modulation order $M$ to be a power of two such that we can define its binary image. The binary image of $\bm{\lambda}$, $\bm{bi}(\bm{\lambda} )$, is then encoded by a systematic encoder with information block length $k_\mathsf{c}$, code length $n_\mathsf{c}$, and rate $R_\mathsf{c}=\frac{k_\mathsf{c}}{n_\mathsf{c}}$. The code is denoted by $\mathcal{C}$, with $|\mathcal{C}|=2^{k_\mathsf{c}}$. The parity bits at the output of the encoder are mapped to a sequence of eigenvalues $\bm{\lambda}_\mathsf{par}\in \Lambda_\mathsf{par}$ with modulation order $M_\mathsf{par}=|\Lambda_\mathsf{par}|$ and $\Lambda_\mathsf{par}\subseteq\Lambda$ by the block $\bm{s}(\cdot )$ in [Fig.]{} such that they are uniformly distributed.

Assuming that a high code rate $R_\mathsf{c}$ is used, we accept a small penalty with respect to the optimal channel input distribution and transmit $\bm{\lambda}$ and $\bm{\lambda}_\mathsf{par}$ via time-sharing. The major difference of compared to is the fact that the channel input distribution is not the optimal distribution due to the time-sharing with the sequence $\bm{\lambda}_\mathsf{par}$. Consequently, this causes a performance degradation. However, is highly flexible as the spectral 