Large diameter RFID antenna

I'm trying to do a project based on this instructible:

https://www.instructables.com/id/RFID-cat-door/step3/Make-an-antenna/

We have a new puppy and it likes to go down stairs and eat the cat food, and I want a way to dissuade it from going down there.

I followed the instructions on the instructible, but possibly because my ring is a different size, or that I couldn't get 24AWG wire and so went with 26AWG my antenna isn't registering the tag when I pass it through the center.

I tried looking up ways to calculate what kind of antenna I need, but so far no luck figuring it out, and the instructible author hasn't replied on how they calculated their antenna.

I am tempted to try just wrapping one loop at a time and then trying for a reading, but that seems tedious and possibly error prone.

Has anyone had any luck doing anything like this?