I used a new soil-less method for my recent raised vegetable bed. This is the method I used: Softened the soil by loosening with a fork. Laid soaked cardboard in the diameter of the bed, overlapping each piece. Then covered with a bed of straw, garden compost, wetting each layer. I laid two layers and used peat moss as the last layer, with a finished height of approx 8" and a bed size of 4x6. I planted corn seeds and covered with mesh. The corn is growing, however, the bed is also covered with "grass". What is the best way to get rid of the grass?
I'd be worried that the cardboard will act as a root blocker and the corn won't be able to develop the deep roots it needs.
As for the grass...like any weed, you'd need to pull it out. Corn is a grass too, so unless you planted round-up ready corn, you likely don't want to use any grass poisons (not that you'd want to use poisons on your garden anyways).
Once the grass is pulled out, you can put down a layer of weedblock. I typically stake in kraft paper. You could also use a layer of newspaper.
Sweet corn needs something like 1 foot of soil depth within the first couple weeks of its growth, see here. This requirement goes up to several feet over the lifetime of the corn. I expect your corn will not be able to grow very well in the conditions you described. Perhaps you can puncture your cardboard layer to allow the roots to get at more soil. Pulling weeds is the best approach to getting rid of them.
Is your grass growing from the bed itself, or is it spreading on runners from the edges? And what kind of grass is it?
If you have Bermuda grass, for example, it spreads on runners above ground, as well as below ground roots, so your cardboard wouldn't have worked that well.
There are some herbicides that will kill certain grasses, but not others. I'm not aware of any that specifically say that don't kill corn though.
Applying a thick layer of mulch may help, and also has other benefits, like increased moisture retention. You could lay down newspaper to smother the grass, and then pile wood chips or straw on top of the news paper, to hold in moisture, keep the newspaper from blowing away, and also look nicer.
One thing I'm trying this year is landscaping fabric. You can roll it out on top of your plants, cut holes in it for the plants, and then cover it in mulch. It lets moisture in, but blocks weeds. It also lasts much longer than newspaper or organic mulch, which can be both good and bad. It also costs more than discarded newspaper.