Building separate grounding systems very often leaves you with a potentially dangerous situation...
If the antenna system is connected to an isolated ground rod, the resistance of the soil between your electrical service ground and the antenna system rod will allow high voltage differences to develop between the antenna system parts and the rest of the grounded items in the home.
If a nearby lighting strike induces a fault current of 500 amps in the antenna ground system and the ground rod resistance is only 2 ohms (an excellent and almost impossible value to achieve value in practice), Ohms law says you would expect to see 1000 volts difference between the antenna system and your TV chassis if it's connected to an outlet.
Please, bond the antenna system to the electrical service ground. This is the only way to achieve a connection with a resistance low enough to truly protect you and your property.
The NEC calls for #10 AWG copper at a minimum for the connection from the mast to ground, and the same for the coax grounding block to ground. Heavier gauge wire is even better given it's lower resistance.
If you add a new ground rod, the NEC calls for it to be bonded to your existing electrical service grounding system. The smallest diameter wire allowed by the NEC for this purpose is #6 AWG copper.
http://forum.tvfool.com/showthread.php?t=901
The ultimate purpose of 'grounding' is not to simply connect to earth... But rather, to prevent large voltage differences from developing between objects that could become energized due to a fault. The connection to earth is only one element in the overall method and plan. Properly sized conductors and reliable connections are vital parts of the overall plan to ensure than hazardous voltages differences can be prevented while safely conducting reasonably anticipated fault current(s).