Let $\mathcal{X}$ be an algebraic curve of genus $g\ge 2$ defined over a field ${\mathbb{F}}_q$ of characteristic $p>0$. From $\mathcal{X}$, under certain conditions, we can construct an algebraic geometry code $C_{\mathcal{X}}$. When this code (or its dual) is self-orthogonal under the symplectic product, a quantum algebraic geometry code $Q_{\mathcal{X}}$ is constructed. In this paper we study the construction of such codes from curves with automorphisms and the relation between the automorphism group of the curve $\mathcal{X}$ and the codes $C_{\mathcal{X}}$ and $Q_{\mathcal{X}}$.

The main purpose of this paper is to provide an asymptotically optimal test. The proposed statistic is of Neyman-Pearson-type when the parameters are estimated with a particular kind of estimators. It is shown that the proposed estimators enables us to achieve this end. Two particular cases, $AR\left(1\right)$ and $ARCH$ models were studied and the asymptotic power function was derived.