Hermitian codes from higher degree places

Abstract

Matthews and Michel \cite{Michel} investigated the minimum distances of certain algebraic-geometry codes arising from a higher degree place $P$. In terms of the Weierstrass gap sequence at $P$, they proved a bound that gives an improvement on the designed minimum distance. In this paper, we consider those of such codes which are constructed from the Hermitian function field $\mathbb F_{q^2}(\HC)$. We determine the Weierstrass gap sequence $G(P)$ where $P$ is a degree $3$ place of $\mathbb F_{q^2}(\HC)$, and compute the Matthews and Michel bound with the corresponding improvement. We show more improvements using a different approach based on geometry. We also compare our results with the true values of the minimum distances of Hermitian $1$-point codes, as well as with estimates due to Xing and Chen \cite{XC}

Multi-Point Codes From Generalized Hermitian Curves

“…Fix r = 0, we consider the one-point code C 0,s over F 8 which is dual to C 3,32−s up to equivalence. Counting the lattice points in both sets Ω 0,s and Ω 3,s , we find the Weierstrass sets H 0 = {0, 4,7,8,9,11,12,13,14, . .…”

On the dual minimum distance and minimum weight of codes from a quotient of the Hermitian curve

“…S. Park gave explicit formulas for the dual minimum distance of such codes (see [15]). More recently, Hermitian codes from higher-degree places have been considered in [11]. The dual minimum distance of many three-point codes on the Hermitian curve is computed in [1], by extending a recent and powerful approach by A. Couvreur (see [5]).…”

Problems and results in

“…Give a bound on the maximum number N(d) of common points in P G(2, q 2 ) of H with F where F ranges over F d (m 0 .m 0 , m 0 ). Problem 7.1 is of interest in the study of algebraic geometric codes on the Hermitian curves arising from a higher degree place, see [17].…”