Multi-point Codes from the GGS Curves

TL;DR

This paper constructs and analyzes algebraic geometric codes from GGS curves, providing explicit descriptions of Riemann-Roch spaces, Weierstrass semigroups, and pure gaps, leading to the discovery of multi-point codes with record parameters.

Contribution

It explicitly characterizes Weierstrass semigroups and pure gaps for GGS curves, enabling the construction of multi-point AG codes with improved parameters.

Findings

A new record code with parameters [216,190,≥18] over GF(64)

Explicit bases for Riemann-Roch spaces on GGS curves

Characterization of Weierstrass semigroups and pure gaps

Abstract

This paper is concerned with the construction of algebraic geometric codes defined from GGS curves. It is of significant use to describe bases for the Riemann-Roch spaces associated with totally ramified places, which enables us to study multi-point AG codes. Along this line, we characterize explicitly the Weierstrass semigroups and pure gaps. Additionally, we determine the floor of a certain type of divisor and investigate the properties of AG codes from GGS curves. Finally, we apply these results to find multi-point codes with excellent parameters. As one of the examples, a presented code with parameters $ [216,190,\geqslant 18] $ over $ \mathbb{F}_{64} $ yields a new record.