In mobile communication systems such as LTE, Hybrid-ARQ (HARQ) schemes based on Turbo codes have proved to be very effective in ensuring reliable transmission. However, several modified HARQ and Turbo decoding strategies have been recently developed to further improve performance. This paper proposes an enhanced coding scheme for LTE which couples a modified Turbo coding strategy with Reliability Based Hybrid ARQ (RB-HARQ). The modified LTE Turbo codes employs an asymmetric encoder and uses a different channel reliability estimate during iterative decoding along with the estimated noise variance. Moreover, an adaptive extrinsic scaling factor is incorporated in the decoder to enhance its performance and provide a stopping criterion. Simulation results show that the proposed modified asymmetric LTE Turbo codes provides an average gain of 1 dB in Eb/N0 over a conventional LTE Turbo code with RB-HARQ and also improves the throughput.