  We have studied the masses for fully open-flavor tetraquark states
$bc\bar{q}\bar{s}$ and $sc\bar{q}\bar{b}$ with quantum numbers
$J^{P}=0^{+},1^{+}$. We systematically construct all diquark-antiquark
interpolating currents and calculate the two-point correlation functions and
spectral densities in the framework of QCD sum rule method. Our calculations
show that the masses are about $7.1-7.2$ GeV for the $bc\bar{q}\bar{s}$
tetraquark states and $7.0-7.1$ GeV for the $sc\bar{q}\bar{b}$ tetraquarks. The
masses of $bc\bar{q}\bar{s}$ tetraquarks are below the thresholds of
$\bar{B}_{s}D$ and $\bar{B}_{s}^{*}D$ final states for the scalar and
axial-vector channels respectively. The $sc\bar{q}\bar{b}$ tetraquark states
with $J^{P}=1^{+}$ lie below the $B_{c}^{+}K^{*}$ and $B_{s}^{*}D$ thresholds.
Such low masses for these possible tetraquark states indicate that they can
only decay via weak interaction and thus are very narrow and stable.
