  Relativistic supernovae constitute a sub-class of type Ic supernovae (SNe).
Their non-thermal, radio emission differs notably from that of regular type Ic
supernovae as they have a fast expansion speed (with velocities $\sim$ 0.6-0.8
c) which can not be explained by a "standard", spherical SN explosion but
advocates for a quickly evolving, mildly relativistic ejecta associated with
the SN. In this paper, we compute the synchrotron radiation emitted by the
cocoon of a long gamma-ray burst jet (GRB). We show that the energy and
velocity of the expanding cocoon, and the radio non-thermal light curves and
spectra are consistent with those observed in relativistic SNe. Thus, the radio
emission from this events is not coming from the SN shock front, but from the
mildly relativistic cocoon produced by the passage of a GRB jet through the
progenitor star. We also show that the cocoon radio emission dominates the GRB
emission at early times for GRBs seen off-axis, and the flux can be larger at
late times compared with on-axis GRBs if the cocoon energy is at least
comparable with respect to the GRB energy.
