Well how secure you need it to be depends on a lot of things. How much predator pressure are you dealing with in the daytime (it's considerably lower in my backyard than in an empty field across town, for example) so basically where do you live?
Also, will they be locked into the secure coop at dusk? Or does the run need to be fort knox for overnight protection as well?
Mine is built of cattle panels, with welded wire on top of that to make the holes smaller, and a 1ft welded wire dig-prevention skirt all around. It's fairly secure from foxes, dogs, raccoons, possums and skunks. However, if there were minks or other critters they can get in MUCH smaller holes (1/2" I think I read) it would not keep them out ---- that would factor into your fencing requirements.
I've dispatched 10 raccoons and 2 foxes in less than 4 months, and lost 6 free ranged roosters to dogs. It's a tough world out there where they live now.