Forward error correction codes are commonly adopted in dual-hop relay communications to ensure the link-layer communication reliability. Among all FEC codes, Luby Transform (LT) codes are favorable because of their low decoder complexity and rate adaptability to channel dynamic fading. To alleviate the high computation cost in the primitive LT-based cooperative communications, hybrid decomposed LT (h-DLT) codes are proposed in our recent work. Theoretical analyses show that reduced energy consumption and latency can be achieved for h-DLT codes assisted cooperative relay communications. In this paper, the practical implementation of h-DLT codes in cooperative relay communications with limited storage capability is investigated. As the direct application of the original h-DLT codes to practical systems can induce high communication costs, we first design a new type of h-DLT codes that enables energy-efficient reliable cooperative communications. Then, the code construction algorithms are provided and the corresponding communications protocol is devised. Finally, simulations are conducted to manifest the performance and benefits of the cooperative relay communication system with the newly proposed h-DLT codes and the effects of multiple design factors, such as the storage size at relays, the mode ratio, the number of relays, and etc.