A high-rate coding scheme that polar codes are concatenated with low density generator matrix (LDGM) codes is proposed in this paper. The scheme, referred to as polar-LDGM (PLG) codes, can boost the convergence speed of polar codes and eliminate the error floor behavior of LDGM codes significantly, while retaining the low encoding and decoding complexity. With a sensibly designed Gaussian approximation (GA), we can accurately predict the theoretical performance of PLG codes. The numerical results show that PLG codes have the potential to approach the capacity limit and avoid error floors effectively. Moreover, the encoding complexity is lower than the existing LDPC coded system. This motives the application of powerful PLG codes to satellite communications in which message transmission must be extremely reliable. Therefore, an adaptive relaying protocol (ARP) based on PLG codes for the relay satellite system is proposed. In ARP, the relay transmission is selectively switched to match the channel conditions, which are determined by an error detector. If no errors are detected, the relay satellite in cooperation with the source satellite only needs to forward a portion of the decoded message to the destination satellite. It is proved that the proposed scheme can remarkably improve the error probability performance. Simulation results illustrate the advantages of the proposed scheme