shape, they have some other advantages. For instance, the first-order soliton transmission does not require compensation or as dispersion and nonlinearity are balanced and hence compensated. This work attempts to approach the limits of current NFT-based systems. To improve the spectral efficiency further, one should use higher-order solitons as well as the continuous part of the nonlinear spectrum together [@aref2016ecoc]. However, the channel equalization will not be as easy as the one for the first-order solitons and the channel model is not yet fully known.

The remainder of the paper is organized as follows. In [Section ]{}, we describe pulse propagation in an optical fiber and the -based transmission scheme. In [Section ]{}, we optimize the input distribution and in [Section ]{}, we introduce and describe the proposed scheme and derive an achievable rate. In [Section ]{}, we present numerical results for , both from Monte-Carlo simulation and simulation, and in [Section ]{} we draw some conclusions.

Notation: The following notation is used throughout the paper. ${\mathfrak{R}\{\cdot\}}$ and ${\mathfrak{I}\{\cdot\}}$ denote the real and the imaginary part of a complex number, respectively, and $\jmath=\sqrt{-1}$ denotes the imaginary unit. Vectors are typeset in bold, e.g., $\bm{x}$, are capitalized, e.g., $X$, and hence 