A punishment zone was an area on Rubicun III where criminal violations resulted in execution. Nobody except the Mediators knew the location of the punishment zone and how long it will be enforced for.
The random and secret location of the zone was intended to serve as a deterrent by creating the ever-present possibility of execution for even the smallest crimes. Across the planet there was only one punishment zone per day and justice for any crime committed within the zone was administered by the Mediators.
In 2364, Wesley Crusher, visiting the planet with the crew of the USS Enterprise-D, violated Edo law by disturbing new plants within a punishment zone. The Mediators accused him of the crime and would have executed him if not for the intervention of Captain Jean-Luc Picard. (TNG: "Justice")