I asked the same question earlier (cuz I don't want to dig a 12" trench around the entire run and coop).
Somebody suggested to attach hardware cloth around the bottom of the OUTSIDE of the run.
The cleaver part is to have the hardware cloth extend out and away from the run about a foot or so and secure it with J-pins (or whatever).
That way, the predators walk up to the run and start pawing at it, then digging, but all they're able to scratch at is the hardware cloth - not the actual dirt!
And as smart as the critters may be, they aren't smart enough to back up about a foot and start digging there.
He also said to cover it with dirt or sod so that it won't be such a tripping hazzard.
I can't attest to how well this works, but it sounds pretty solid to me.
I'm gonna do it!