Network coding [1] is a technique to maximize communication rates within a network, which may be used to devise communication protocols for simultaneous multi-party transmission of information. Linear network codes are examples of such protocols in which the local computations performed at the nodes in the network are limited to linear transformations of their input data, represented as elements of a ring such as the integers modulo 2. We demonstrate that the quantum linear network coding protocols of Kobayashi et al. [17, 18], which coherently simulate classical linear network coding protocols, correspond in a natural way to measurement-based quantum computations with graph states over qudits [21, 4, 8] having a structure directly related to the network.