It turns out the codes distributed weren't matched up to a specific Origin account; they could be used by anyone. Oh, and the codes worked multiple times, which means users could download as many under-$20 games as they wanted. Oh, and one more thing: Reddit figured all of this out.
As you can imagine, all hell broke loose on Origin, as what must have been thousands of users downloaded games like Dead Space 2, Burnout Paradise, Sim City 4, and Mass Effect 2 for exactly zero dollars. The site went down, and when it came back up the codes (which were supposed to work until October 21) had been disabled. Not only that, but all promo codes have been disabled for the time being, apparently.
There's no word on if EA will still honor the codes from the survey [see update: They will], or if there will be a new system in place at all, but we can almost guarantee that whatever solution EA comes up with, the IT team will probably have double- and triple-checked it for security, just to be sure. We've contacted EA for comment on what its plans are, and will let you know if we hear anything more.
Update: EA has gotten back in touch to say that those users who completed the survey will be getting new codes via email soon, to provide the specified discount. In other words, no more stealing allowed.