Optical eigenvalue transmission based on inverse scattering transform (IST) has been studied for one of approaches to overcome the Kerr nonlinearity limit in fiber optic communications. In the recent decade, several multilevel modulation schemes that are based on IST, such as 16-ary and 64-ary signals using on-off encoding and b-modulation of multieigenvalues, have been proposed. To increase the transmission capacity and extend the transmission distance, applications of machine learning-based approaches to IST-based transmission have been proposed and demonstrated. Another prospective approach to increase the transmission capacity and extend the transmission distance in fiber optic communication involves the application of soft-decision forward error correction (SD-FEC). However, the applicability of SD-FEC to eigenvalue-modulated signals has not yet been investigated in detail because the distribution of the received signal is complicated for IST-based transmissions. In this paper, we describe in detail the theory of optical eigenvalue transmission, including the design of multilevel eigenvalue-modulated signals and the effects of noise and scaring parameters (coefficients for normalization of the nonlinear Schrödinger equation). We explain why neural network (NN) demodulators are advantageous for eigenvalue transmission systems. Consequently, we propose a combination of NN-based demodulators and SD-FEC decoding. A multilabel NN-based demodulator is employed to compute the L-value from the received eigenvalue pattern at the receiver. For a 16-ary eigenvalue-modulated signal, the proposed method outperformed a combination of the Gaussian approximation and SD-FEC in the simulation. Moreover, the experimental results show successful operation with error-free transmission through a 3000-km optical fiber line. In addition, we experimentally demonstrate the applicability of SD-FEC to a 4096-ary eigenvalue-modulated signal. The experimental results indicate that an achievable transmission distance can be extended to 1200 km using the NN demodulator and SD-FEC.