The antenna is nothing more than a coil of magnet wire connected to the RF reader. Most RF readers come with an antenna, but some allow the use of an external antenna. For this project I made a square antenna of 10 x 10 inches by winding 24 gauge magnet wire 24 turns around a sawed off bucket to make it sturdy. The distance between turns has to be as small as possible. I mounted the thing on the plywood with two pieces of wood as spacers to make it stand off from the wall a bit so the cat could activate it farther from the wall. The
35mm disc tags that I use are read up to 4 inches of the plane of the coil. The RF reader and antenna are powered with the 5v from the Arduino. Even though the Arduino works fine with just the USB cable, the RF reader works better when the Arduino is plugged into the wall with the
9v power adapter.
More info about coil antennas:
Microchip Inc. Antenna circuit design for RFID applications (pdf)