  Ballico proved that a smooth projective variety $X$ of degree $d$ over a
finite field of $q$ elements admits a smooth hyperplane section if $q\geq
d(d-1)^{\dim X}$. In this paper, we refine this criterion for higher
codimensional linear sections on smooth hypersurfaces and for hyperplane
sections on Frobenius classical hypersurfaces. We also prove a similar result
for the existence of reduced hyperplane sections on reduced hypersurfaces.
