  We study finite (physical) volume and scaling violation effects of the Landau
gauge ghost propagator as well as of the running coupling $\alpha_s(p)$ in the
$SU(2)$ lattice gauge theory. We consider lattices with physical linear sizes
between $aL \simeq 3$ fm and $7$ fm and values of lattice spacing between
$a=0.2$ fm and $0.07$ fm. To fix the gauge we apply an efficient gauge fixing
method aimed at finding extrema as close as possible to the global maximum of
the gauge functional. We find finite volume effects to be small for the lattice
size $aL \simeq 3$ fm at momenta $|p|\, \ge \, 0.6$ GeV. For the same lattice
size we study extrapolations to the continuum limit of the ghost dressing
function as well as for the running coupling with momenta chosen between $|p| =
0.41$ GeV and $3.2$ GeV. We present fit formulae for the continuum limit of
both observables in this momentum range. Our results testify in favor of the
decoupling behavior in the infrared limit.
