The 23" is for a 1/4 wave antenna at the frequency for the aviation radio
band (123 MHz). The transponder operates at about 1,000 MHz, so a 1/4 wave
is much smaller.

You do not even need to do much calculation, since the radius of the ground
plane should be equal to the height of the antenna. So, the diameter would
be twice the antenna height.

The shape need not be round. It can be square or rectangular. Generlaly, a
slight improvement will be achieved with a larger ground plane.

Also remember, that, even though the transmitter is pulsed at a very short
duration, there could be some possible danger having it close to your body.
If you use the ground plane to "hide" the antenna from you, there would be
little danger from exposure. The jury is still out on this one. I do not
think there is any credible evidence of a danger, but I would not expose
myself unless there was no other choice. Mine is mounted under my seat,
upside down.

Upside down is better because radio waves at this frequency are "line of
sight" and the ground stations interrogating you are all below you.

Do a good job of coax installation. I have seen some crappy jobs. Also
remember that coax attenuation at 1,000 MHz with RG-58 is very high - so
keep the run short.

Colin Lamb