** I was thinking of running the drain hose up to the 20 inch mark and then down through the floor **
That should allow the dishwasher to function properly. In fact, most new dishwashers have their drain hose taped to the proper height on the side of the cabinet with a big "DO NOT TOUCH" sign across it. Some people still manage to screw it up though. Anyway...
As tomh said, some local code require an air gap to be installed with all new installations which should be done if required in your area.
[i]The Appliance Information Site[/i]