An upper bound for the genus of a curve without points of small degree

In this paper I prove that for any prime $p$ there is a constant $C_p>0$ such that for any $n>0$ and for any $p$-power $q$ there is a smooth, projective, absolutely irreducible curve over ${\mathbb{F}}_q$ of genus $g\le C_p{q}^n$ without points of degree smaller than $n$.