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This is really neat!

A question I had about QR codes:

Let’s say I want to produce a QR code that is as close as possible to an image (let’s say the Nike swoosh logo), if I just added that to the center and added the 3 orientation markers on the corners it wouldn’t be a proper QR code yet, my question is what is the minimal amount of edits I would need to do from that symbol to get a valid QR code?

I’m thinking of something like people buying and hoarding domains in early 00’s (even now), if I could generate the minimal QR code (minimal is important here in the same way that .com is treated as superior to .ca or .net) of well-known symbols and buy the resulting domain it points to (if it even points to a domain) couldn’t I sell this to companies with the associated domain as a re-direct as well?

From another comment:

https://research.swtch.com/qart

Looks like this is definitely possible, the results aren't as clean as you might hope though. Maybe they could be improved by introducing some intentional errors (which were explicitly avoided here).

Using their logo in this way is probably a trademark violation, so they would be able to get your QR code “for free” (minus the costs involved with consulting their legal division).

You could maybe determine this using a SAT solver. You would have to encode the entire QR code specification. I was thinking of doing this to find a minimalist QR code that just had as few black cells as possible.