I'm a bit unclear on how the "quads" are chosen (ie. which 4 stars are grouped together). I assume your image and the sky-survey database you're comparing to are at a completely different detail-level. Choosing every combination of 4 stars seems not practical. If you just choose groups of 4 that are next to each other then you may find the database has dozens of stars in between your quad's stars (b/c it's at a higher detail level)

The slides acknowledge this by saying " Finally, there is the additional problem of distractor & dropout stars", but I don't see their solution :)

Maybe this is a clue "Cycle through all possible valid* quads (brightest first) and compute their corresponding codes" but there are no details.