lawn grass invading raised bed.. help!

I built a raised bed next to my lawn a few years ago. I wasn't thinking a lot when I did it. I just set the boards on the ground. I also added brick edging that is about an inch deep in the ground. I have the kind of grass in my lawn that sends out runners and puts down roots at the runner joint. I think it also can go underground and is going under and between the bricks. In any event, it's getting into the bed by going under and up the boards.
I'm thinking of digging out inside the boards and putting down some corrugated plastic I have. It would be a bit of work, so I don't want to repeat it. Any idea how deep I have to go to stop the grass? I don't want to work any harder than I have to! Living in Southern California I can get started this month, but need the motivation of knowing it would be worthwhile...
thoughts? Anyone have similar experience with invasive lawn grass (the kind I'm describing which isn't really traditional grass but is sure super drought tolerant, which is important right now).
Thanks!

Texas has nothing but grasses that do that. It depends on how deep you need to dig- but those seeds that get thrown over the boards/bricks are from above. So keep in mind the grass you have and the depth it runs.

Thanks. I don't mind that weeds (or grass) that sprout at the top. It's easy enough to pull those weeds out. It's the grass roots/runners that are coming up from below that are coming from under the side bricks/boards. You can pull off the top of the grass, but it just keeps re-sprouting from the darn runners. So I'd like to make a barrier between the lawn and the bed, but I'm not sure how deep the barrier has to go. I saw somewhere else a person mentioned 30" deep. That's much deeper than I think I am willing to go or can go. I was wondering if 6" would do it. I'm not sure if the grass is sending runners or roots, or how deep they can go to get around. Right now they just go under the 1" deep bricks I think.

If you have coastal/bermuda, 8" is a better depth, 36" for bamboos, Johnson grass; besides 8" below and 4" above makes a mole barrier as well. Calif grasses, hmmm, not sure of your running grass types, chuckl

In addition to a dug barrier, we set in (the ground) a border of flat pavers (8" wide, 2" deep, any length) along the outer perimeter of all of our gardens, which eliminates having to "trim" the grass around raised beds and walls. We just mow over top of the flat concrete border!