Step 7: Optional: Build your own RF antenna
Guides for the construction of antennas are available for download on the web (e.g. http://ww1.microchip.com/downloads/en/devicedoc/51115f.pdf). They help quite a bit, but I still ended up just trying a bunch of things to see what worked best.
An antenna coil is as simple as this: http://img101.imageshack.us/i/coil2sm.jpg/, just a wire wound up a bunch of turns with the two ends connected to the RF reader.
I used 24 Ga magnet wire and a 1.25" diameter tag to test the antenna.
The following worked well (range is in distance from the plane of the coil):
1.6" diameter, 90-110 turns, no range measurement
3.1" diameter, 75 turns, no range measurement
4.1" diameter, 48 turns, no range measurement
6" diameter, 43 turns, range 3-3.5"
9" diameter, 35 turns, range 4"
10" square shape, 31 turns, range 1"
So the largest range was with a 9" diameter coil. Larger than that, the range first went down to zero and then disappeared from the center. If you draw power from the Arduino, use the 9v adapter, not USB only, because the latter provides less power. Also important is when the coil is complete, the wire should be guided away from the coil.