  The radius and surface composition of an exploding massive star, as well as
the explosion energy per unit mass, can be measured using early ultraviolet
(UV) observations of core-collapse supernovae (CC SNe). We present the results
from a simultaneous \GALEX and Palomar Transient Factory (PTF) search for early
UV emission from SNe. We analyze five CC SNe for which we obtained $NUV$
measurements before the first ground-based $R$-band detection. We introduce
SOPRANOS, a new maximum likelihood fitting tool for models with variable
temporal validity windows, and use it to fit the \citet{SapirWaxman2017} shock
cooling model to the data. We report four Type II SNe with progenitor radii in
the range of $R_*\approx600-1100R_\odot$ and a shock velocity parameter in the
range of $v_{s*}\approx 2700-6000 \,\rm km\,s^{-1}$
($E/M\approx2-8\times10^{50}\,\rm erg/M_\odot$) and one type IIb SN with
$R_*\approx210R_\odot$ and $v_{s*}\approx11000 \rm\, km\,s^{-1}$
($E/M\approx1.8\times10^{51}\,\rm erg/M_\odot$). Our pilot GALEX/PTF project
thus suggests that a dedicated, systematic SN survey in the $NUV$ band, such as
the wide-field UV explorer \textit{ULTRASAT} mission, is a compelling method to
study the properties of SN progenitors and SN energetics.
