Sending a new code every time sounds like what the pretty old and mundane rolling code openers have done for years. When you programmed the remote to the Homelink did you just press and hold the button on the opener's remote or did you do the press-release-press cycle while you held the Homelink buttons?
If you did the press-release-press and it still doesn't work you might need this bridge kit to make it work:
https://secure.homelink.com/homelink...ter-kit-detail