Long duration gamma-ray bursts are a rare subclass of stripped-envelope core-collapse supernovae that launch collimated relativistic outflows (jets). All gamma-ray-burst-associated supernovae are spectroscopically of Type Ic with broad lines, but the fraction of broad-lined Type Ic supernovae harboring low-luminosity gamma-ray-burst remains largely unconstrained. Some supernovae should be accompanied by off-axis $gamma$-ray burst jets that remain invisible initially, but then emerge as strong radio sources (as the jets decelerate). However, this critical prediction of the jet model for gamma-ray bursts has yet to be verified observationally. Here, we present K. G. Jansky Very Large Array observations of 15 broad-lined supernovae of Type Ic discovered by the Palomar Transient Factory in an untargeted manner. Most of the supernovae in our sample exclude radio emission observationally similar to that of the radio-loud, relativistic SN,1998bw. We constrain the fraction of 1998bw-like broad-lined Type Ic supernovae to be <~ 41% (99.865% confidence). Most of the events in our sample also exclude off-axis jets similar to GRB 031203 and GRB 030329, but we cannot rule out off-axis gamma-ray-bursts expanding in a low-density wind environment. Three supernovae in our sample are detected in the radio. PTF11qcj and PTF14dby show late-time radio emission with average ejecta speeds of ~(0.3-0.4)c, on the dividing line between relativistic and ordinary supernovae. The speed of PTF11cmh radio ejecta is poorly constrained. We estimate that <~ 85% (99.865% confidence) of the broad-lined Type Ic supernovae in our sample may harbor off-axis $gamma$-ray-bursts expanding in media with densities in the range probed by this study.
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