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S. Tidal. To go home, make sure you have switched off the Birth Island code and enter the S.S. Tidal. It can be found in the south part of the island. You don't need a ticket to go home, but you will find yourself in Lilycove City, which is dangerous for trainers with low-level Pokemon. When you've completed all the steps and have a Deoxys, you want to make sure you save the game so you don't lose your progress.
Board the S. Save the game.