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 $v_{s*}approx 2700-6000 ,rm km,s^{-1}$ ($E/Mapprox2-8times10^{50},rm erg/M_odot$) and one type IIb SN with $R_*approx210R_odot$ and $v_{s*}approx11000 rm, km,s^{-1}$ ($E/Mapprox1.8times10^{51},rm erg/M_odot$). 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.
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