Scale dependence of top-quark cross section at $e^{+}{e}^{-}$ colliders near production threshold at NNNLO

The top-quark threshold cross section at $e^{+}{e}^{-}$ colliders near production threshold is investigated. We study the scale dependence of the cross section for $\sigma (e^{+}{e}^{-}\to ttX$) near $\sqrt{s}\simeq 2m_t$ and discuss the theoretical accuracy of the NNNLO prediction. We report that a threshold scan at an $e^{+}{e}^{-}$ collider would allow a determination of the top-quark mass with an accuracy of order 30 MeV.