Cyclic AG-Codes on the Hermitian Curve

TL;DR

This paper constructs cyclic algebraic geometric codes on the Hermitian curve with specific divisors and intersection properties, expanding the understanding of code structures on algebraic curves.

Contribution

It introduces a new class of cyclic AG-codes on the Hermitian curve with divisors supported on intersections with chords, leveraging automorphism group actions.

Findings

Codes are constructed with divisors supported on intersection points with chords.

The divisor D consists of points in a single orbit under a cyclic automorphism group.

The codes have specific algebraic and geometric properties related to the Hermitian curve.

Abstract

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 $\mathrm{Aut}(H_q) = \mathrm{PGU}(3,q)$.