  Analogous to the well-known proton charge radius puzzle, a similar puzzle
exists for the deuteron charge radius, $r_{d}$. There are discrepancies
observed in the results of $r_{d}$, measured from electron-deuteron ($e-d$)
scattering experiments, as well as from atomic spectroscopy. In order to help
resolve the charge radius puzzle of the deuteron, the PRad collaboration at
Jefferson Lab has proposed an experiment for measuring $r_{d}$, named DRad.
This experiment is designed to measure the unpolarized elastic $e-d$ scattering
cross section in a low-$Q^{2}$ region. To extract the cross section with a high
precision, having reliable knowledge of QED radiative corrections is important.
In this paper, we present complete numerical calculations of the lowest-order
radiative corrections in $e-d$ scattering for the DRad kinematics. The
calculations have been performed within a covariant formalism and beyond the
ultra-relativistic approximation ($m_{e}^{2} \ll Q^{2}$). Besides, we present a
systematic uncertainty on $r_{d}$ arising from higher-order radiative
corrections, estimated based on our cross-section results.
