scheme via discrete-time Monte-Carlo and simulations. For the mapping $\bm{bi}(\cdot)$ (see [Fig.]{}), we use Gray labeling. Also, for the , we use the binary codes of the standard with code length $n_\mathsf{c}=64800$ and code rates $R_\mathsf{c}=\{$$1/4$, $1/3$, $2/5$, $1/2$, $3/5$, $2/3$, $3/4$, $4/5$, $5/6$, $8/9$, $9/10\}$. For the parity symbols, we use the constellation arising from [Algorithm]{}, depicted in [Fig.]{}.

Detection {#sec:detection}
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For the simulation, we simulate a continuous signal and hence, we require a detector. We use the following method to deal with the variable pulse durations: We set a threshold $\theta$ sufficiently higher than the noise. Once the magnitude of the signal rises above $\theta$, we save the time as $t_\mathsf{start}$ and when the magnitude of the signal falls below $\theta$, we save the time as $t_\mathsf{end}$. We then extend the interval bounded by $t_\mathsf{start}$ and $t_\mathsf{end}$, i.e., $\tilde{t}_\mathsf{start} = t_\mathsf{start} - \delta_t$ and $\tilde{t}_\mathsf{end} = t_\mathsf{end} + \delta_t$. Calculating the over the interval $[\tilde{t}_\mathsf{start}, \tilde{t}_\mathsf{end}]$ using the spectral method [@yousefi2014nftI-III Part II, Section IV] and only considering the imaginary part of the discrete eigenvalue gives the received symbol $y$. This approach requires that the is sufficiently high. As the model has the same requirement due to 