Messy cats: Cat box woes?

I use a very deep plastic storage box without the cover (at least a foot high). Most litter boxes are much too short for cats that do this, or they have a cover and are fairly expensive. I cut part of the end off so it's easier for my cat to jump over. To make it harder for the litter to end up over the side, I put cardboard on the side away from the wall. She can dig and dig and the litter doesn't go anywhere I don't want it to.