  An irreducible smooth projective curve over $\mathbb{F}\_q$ is called
\textit{pointless} if it has no $\mathbb{F}\_q$-rational points. In this paper
we study the lower existence bound on the genus of such a curve over a fixed
finite field $\mathbb{F}\_q$. Using some explicit constructions of
hyperelliptic curves, we establish two new bounds that depend linearly on the
number $q$. In the case of odd characteristic this improves upon a result of R.
Becker and D. Glass. We also provide a similar new bound when $q$ is even.
