Bonding new ground wire from ground rod to exsiting house grounding block
For lightning protection, the antenna and mast should be connected directly to a ground rod. Use straight runs and avoid splices.

Even nearby lightning, as opposed to a direct hit. creates very high magnetic and induced electric fields, so avoid sharp bends in the ground wire and do not run near or parallel to other wiring. 

Ground wire and rods are fairly inexpensive and are easily installed. Note that the size of the ground rod(s) depends on soil conductivity -- check local conditions.