  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.
