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It is anticipated that a large amount of multicast traffic needs to be supported in future communication networks. The network coding technique proposed recently is promising for establishing multicast connections with a significantly lower bandwidth requirement than that of traditional Steiner-tree-based multicast connections. How to design multicast network topologies with the consideration of efficiently supporting multicast by the network coding technique becomes an important issue now. It is notable, however, that the conventional algorithms for network topology design are mainly unicast-oriented, and they cannot be adopted directly for the efficient topology design of network-coding-based multicast networks by simply treating each multicast as multiple unicasts. In this paper, we consider for the first time the novel topology design problem of network-coding-based multicast networks. Based on the characteristics of multicast and network coding, we first formulate this problem as a mixed-integer nonlinear programming problem, which is NP-hard, and then propose two heuristic algorithms for it. The effectiveness of our heuristics is verified through simulation and comparison with the exhaustive search method. We demonstrate in this paper that, in the topology design of multicast networks, adopting the network coding technique to support multicast transmissions can significantly reduce the overall topology cost as compared to conventional unicast-oriented design and the Steiner-tree-based design.