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Forward error correction (FEC) is commonly adopted in cooperative relay communications to ensure link-layer communication reliability. Among those schemes, rateless fountain codes, such as Luby Transform (LT) codes, are favorable for their low complexity and rate adaptability to channel fading dynamics. However, the cooperative transmission schemes based on primitive fountain codes induce either heavy computation cost or large end-to-end latency. To address these issues, we explore decomposed LT (DLT) codes, which comprise of two layers of random encoding but only a single layer of decoding. By implementing the two layers of encoding at the source and the relay(s) respectively, the cooperative system can ensure communication reliability on both source-relay and relay-destination links with reduced computation cost and latency. In this work, we first develop a general decomposition technique for the DLT code construction. Based on this, we further propose a hybrid decomposition algorithm tailored for LT codes with robust Soliton distribution (RSD). The resultant hybrid DLT (h-DLT) codes facilitate flexible computation cost allocation. The h-DLT codes based cooperative relay communication protocol is then developed and analyzed in terms of the transmission latency and energy consumption.