Percolation theory looks at problems such as the probability of being able to traverse some region with random obstacles. It is motivated by problems such as modeling the flow of a fluid in a porous medium.
Here’s a percolation problem for QR codes: What is the probability that there is a path from one side of a QR code to the opposite side? How far across a QR code would you expect to be able to go? For example, the QR code below was generated from my contact information. It’s not possible to go from one side to the other, and the red line shows what I believe is the deepest path into the code from a side.
This could make an interesting programming exercise. A simple version would be to start with a file of bits representing a particular QR code and find the deepest path into the corresponding image.
The next step up would be to generate simplified QR codes, requiring certain bits to be set, such as the patterns in three of the four corners that allow a QR reader to orient itself.
The next step in sophistication would be to implement the actual QR encoding algorithm, including its error correction encoding, then use this to encode random data.
(Because of the error correction used by QR codes, you could scan the image above and your QR reader would ignore the red path. It would even work if a fairly large portion of the image were missing because the error correction introduces a lot of redundancy.)