Non-orthogonal multiple access (NOMA) has the potential to provide higher throughput than conventional orthogonal multiple access (OMA), which has been considered as a key technology for 5G. NOMA in satellite communication system can provide anytime, anywhere access with improved spectral efficiency and system capacity because of its ubiquitous coverage. However, the characteristics of the satellite channels are different from that of the terrestrial network, i.e., huge time delay and Doppler shift. In this paper, different from the existing works, which mainly focus on the performance of NOMA in static terrestrial base stations and Geostationary orbit (GEO) scenarios, we investigate the performance analysis of downlink NOMA in dynamic low earth orbit (LEO) satellite communication system with Doppler shift considered. We combine NOMA and orthogonal frequency division multiplexing (OFDM) for better spectral efficiency. Our channel model includes both small scale model and large scale model. For simplify, we only consider two users in one spot beam. Besides, we express the performance analysis of downlink NOMA in LEO satellite communication system, like ergodic capacity, outage probability (OP) and mutual information. However, in traditional NOMA scheme, the proceeding error decision of the high power user will cause the deterioration of the subsequent detection performance. Therefore, a symmetrical coding (SC) scheme for different modulation mode is proposed for low power users to get better performance. Finally, simulation results validate the performance of the NOMA scheme is better than that of the OMA scheme. The proposed SC scheme can achieve a prominent increase performance contrasted to the traditional NOMA scheme. INDEX TERMS LEO satellite communication system, NOMA, ergodic capacity, outage probability, mutual information, symmetrical coding scheme. AIJUN LIU (Member, IEEE) received the B.S. degree in microwave communications and the M.S. and Ph.D. degrees in communications engineering and information systems from the College of Communications Engineering, Army Engineer