On board 13.56MHz RFID antenna design
Interesting reference documents:


Application notes from NXP, in particular AN1445, AN11535
Training sessions from NXP: NFC Reader Design: Antenna design considerations


They basically say that:


reading distance will be related to antenna diameter,
inductance should be kept in the 0.3-3µH range.


Here you are facing "passive" (in spec terminology) NFC communication. This is done through load modulation. Communication medium is a transformer[1].

Being a transformer, voltage at secondary coil poles is a factor of the number of turns. So the more turns, the more voltage (and the less current). Because we use digital ICs with clamping diodes on secondary side, having more voltage there makes little interest: it will flow through clamping diodes and will basically be useless.

Another important design consideration for a PCB with both antenna and components is that you should not block the magnetic field with the chips and traces. Keep antenna trace at the outer region of board, put components in the middle. 

1: Then talking about antenna is somewhat wrong, but this is the usual terminology, so we'll keep it. "Active" communication, on the other hand, is much more like using antennas -- that's why NFC antenna requirements are different depending on whether you intend to use passive only or both communication methods.