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This paper considers the problems of minimizing the completion time and reducing decoding complexity for relay-aided wireless broadcast. Both network coding and scheduling problems are considered. A deterministic network coding algorithm is designed to select innovative encoding vectors, which is applicable to both the base station and the relay. The proposed algorithm can reduce decoding complexity significantly by selecting sparse encoding vectors. Integrating with the proposed network coding algorithm, a scheduling scheme based on dynamic programming is proposed, which is proved to be optimal in terms of minimizing expected completion time. Considering the size of state space and computational complexity of dynamic programming, a greedy scheduling algorithm is also proposed for practical implementation. Simulation shows that the proposed network coding algorithm and scheduling schemes work very well both in reducing completion time and in decoding complexity.