minimum sampling rate for very short duration signals
Any signal of finite extent (duration) in one domain (time) has infinite extent (bandwidth) in the other (frequency).  The bandwidth of a short pulse above some noise floor can be extremely high, due to the pulse window's transform convolved with the carrier frequency.  

Sharp edges or cutoffs imply very high bandwidths.  Thus in practice, most designed pulses are shaped closer to Gaussian, with much wider tails than the main pulse lobe (in both domains).

Even so, if you know the exact frequency of an unmodulated fragment of a pure sine wave, then in zero noise, 2 or 3 non-aliased points are theoretically enough to solve the equations for the sinewave's  amplitude and phase.