  We examine the bright radio synchrotron counterparts of low-luminosity
gamma-ray bursts (llGRBs) and relativistic supernovae (SNe) and find that they
can be powered by spherical hypernova (HN) explosions. Our results imply that
radio-bright HNe are driven by relativistic jets that are choked deep inside
the progenitor stars or quasi-spherical magnetized winds from fast-rotating
magnetars. We also consider the optical synchrotron counterparts of
radio-bright HNe and show that they can be observed as precursors several days
before the SN peak with an r-band absolute magnitude of M_r ~ -14 mag. While
previous studies suggested that additional trans-relativistic components are
required to power the bright radio emission, we find that they overestimated
the energy budget of the trans-relativistic component by overlooking some
factors related to the minimum energy of non-thermal electrons. If an
additional trans-relativistic component exists, then a much brighter optical
precursor with M_r ~ -20 mag can be expected. Thus, the scenarios of
radio-bright HNe can be distinguished by using optical precursors, which can be
detectable from < 100 Mpc by current SN surveys like the Kiso SN Survey,
Palomar Transient Factory, and Panoramic Survey Telescope & Rapid Response
System.
