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 $G = m(P_2 + \ldots + P_q)$, where $2 \le m \le q-1$ and $\mathrm{supp}(G)$ is the intersection of $H_q$ with a chord $\ell$ minus two points $P_1, P_{q+1}$. The divisor $D = Q_1 + \ldots + Q_{q^2-1}$ consists of all $q^2 - 1$ points in a single orbit under the action of the (cyclic) 2-point stabilizer $\Gamma$ of $(P_1, P_{q+1})$ in $\mathrm{Aut}(H_q) = \mathrm{PGU}(3,q)$.