  Precise predictions for an $e^+e^-\rightarrow t\bar{t}$ cross section are
presented at an energy region from 400 GeV to 800 GeV. Cross sections are
estimated including the beam-polarization effects with full
$\mathcal{O}(\alpha)$, and also with effects of the initial-state photon
emission. A radiator technique is used for the initial-state photon emission up
to two-loop order. A weak correction is defined as the full electroweak
corrections without the initial-state photonic corrections. As a result, it is
obtained that the total cross section of a top quark pair-production receives
the weak corrections of $+4\%$ over the trivial initial state corrections at a
centre of mass energy of 500 GeV. Among the initial state contributions, a
contribution from two-loop diagrams gives less than $0.11\%$ correction over
the one-loop ones at the center of mass energies of from $400$ GeV to $800$
GeV. In addition, an effect of a running coupling constant is also discussed.
