  We construct multipartite graph states whose dimension is the power of a
prime number. This is realized by the finite field, as well as the generalized
controlled-NOT quantum circuit acting on two qudits. We propose the standard
form of graph states up to local unitary transformations and particle
permutations. The form greatly simplifies the classification of graph states as
we illustrate up to five qudits. We also show that some graph states are
multipartite maximally entangled states in the sense that any bipartite of the
system produces a bipartite maximally entangled state. We further prove that
4-partite maximally entangled states exist when the dimension is an odd number
at least three or a multiple of four.
