NEC 810.15 dictates that the mast be grounded.
NEC 810.20(A) dictates that an "antenna discharge unit" (grounding block) be used on the cable.
NEC defines grounding as running wires of a specific minimum size to a "bonded ground" that attaches to your service entrance ground. This may involve multiple ground rods and connecting wires between them.
Similar to this thread, I've gotten conflicting opinions on whether I should/am required to ground the structure the antenna is connected to. Does the structure I attach the antenna to make a difference? (i.e. a mast vs attaching it to something else)