Multi-point codes from the GGS curves

Chuangqiang Hu; Shudi Yang

doi:10.3934/amc.2020020

Article Contents

2020, Volume 14, Issue 2: 279-299. Doi: 10.3934/amc.2020020 open access

This issue Previous Article Locally recoverable codes from algebraic curves with separated variables Next Article Dual-Ouroboros: An improvement of the McNie scheme

Multi-point codes from the GGS curves

Chuangqiang Hu^1, and
Shudi Yang^2, ,

1.
Yau Mathematical Sciences Center, Tsinghua University, Peking, 100084, China
2.
School of Mathematical Sciences, Qufu Normal University, Shandong, 273165, China

^*Corresponding author: Shudi Yang
^*Corresponding author: Shudi Yang

Received: June 2018

Revised: December 2018

Early access: September 2019

Published: May 2020

This work is partially supported by the NSFC (11701317, 11531007, 11571380, 11701320, 61472457) and Tsinghua University startup fund. This work is also partially supported by China Postdoctoral Science Foundation Funded Project (2017M611801), Jiangsu Planned Projects for Postdoctoral Research Funds (1701104C), Guangzhou Science and Technology Program (201607010144) and the Natural Science Foundation of Shandong Province of China (ZR2016AM04)

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Abstract

This paper is concerned with the construction of algebraic-geometric (AG) codes defined from GGS curves. It is of significant use to describe bases for the Riemann-Roch spaces associated with some rational places, which enables us to study multi-point AG codes. Along this line, we characterize explicitly the Weierstrass semigroups and pure gaps by an exhaustive computation for the basis of Riemann-Roch spaces from GGS curves. In addition, we determine the floor of a certain type of divisor and investigate the properties of AG codes. Multi-point codes with excellent parameters are found, among which, a presented code with parameters $ [216,190,\geqslant 18] $ over $ \mathbb{F}_{64} $ yields a new record.

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Mathematics Subject Classification: Primary: 14H55, 11R58, 11T71.

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