Cyclic AG-Codes on the Hermitian Curve

Cyclic AG-codes $C_L(D,G)$ on the Hermitian curve $H_q$ over ${\mathbb{F}}_{q^2}$ are constructed such that $G=m(P_2+\dots +P_q)$, where $2\le m\le q-1$ and $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(G)$ is the intersection of $H_q$ with a chord $\mathrm{\ell }$ minus two points $P_1,P_{q+1}$. The divisor $D=Q_1+\dots +Q_{q^2-1}$ consists of all $q^2-1$ points in a single orbit under the action of the (cyclic) 2-point stabilizer $\mathrm{\Gamma }$ of $(P_1,P_{q+1})$ in $\mathrm{A}\mathrm{u}\mathrm{t}(H_q)=\mathrm{P}\mathrm{G}\mathrm{U}(3,q)$.