A housemate of mine had a very creative solution: as a teacher he said that he would give anybody who cheated a 0 and that appeals could be filed with the Dean's office to prove they did not cheat. Most importantly, nobody would be told if they had been caught cheating, the zero would just show up on their grades.

The system plays on the student's mind: when they submit the first assignment and cheat their motivation to bother to make the minimal effort to dick around with the next one waivers. Most of the cheaters apparently dropped before the midterms. Any whiners with the balls to claim they had not cheated get to make their case to the Dean and voluntarily submit themselves to the school judicial process, but this was generally unneeded as apparently most cheaters dropped the class before midterms. No whiners in your office, and the reviews of you as a teacher are in before you fail them. Seems like it addresses all your problems.