  Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold such that
$\pi_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that
strict upper bounds of 0.07625, 0.1525 and 0.22875 for ${\rm vol}\ {\mathfrak
M}$ imply respective upper bounds of 23, 43 and 79 for $\dim H_1({\mathfrak
M};{\mathbb F}_2 )$. Stronger results hold if we assume that the singular set
$\Sigma$ is a link; specifically, under this assumption, strict upper bounds of
0.305, 0.4575, 0.61, 0.7625 and 0.915 for ${\rm vol}\ {\mathfrak M}$ imply
respective upper bounds of 7, 13, 14, 28 and 29 for ${\rm dim}\ H_1({\mathfrak
M};{\mathbb F}_2 )$. Irreducibility assumptions on the underlying manifold
$|{\mathfrak M}|$ of ${\mathfrak M}$, and of the underlying manifolds of
certain coverings of ${\mathfrak M}$, also give stronger results. The upper
bounds on $\dim H_1({\mathfrak M};{\mathbb F}_2 )$ for an orbifold ${\mathfrak
M}$ whose volume is subject to a suitable upper bound are deduced from upper
bounds on ${\rm dim}\ H_1(|{\mathfrak M}|;{\mathbb F}_2 )$ for an orbifold
${\mathfrak M}$ whose volume is subject to a suitable upper bound.
