On Goppa codes and Weierstrass gaps at several points
type: Item
instance of: scholarly article
title: On Goppa codes and Weierstrass gaps at several points
zbMATH Open document ID: 1070.94029
author: Fernando Torres
author: Cicero Carvalho
published in: Designs, Codes and Cryptography
publication date: 2005-06-15T00:00:00Z
review text: One-point geometric Goppa codes are defined as the algebraic geometric codes \(C(D,G)\) where \(G=\mu Q\) is a multiple of a single point. In [J. Pure Appl. Algebra 84, 199--207 (1993; Zbl 0768.94014)] \textit{A. Garcia}, \textit{S. J. Kim} and \textit{R. Lax} showed that the true minimum distance of these codes can be greater than the Goppa bound for certain values of \(\mu\), and this fact is related to the structure of gaps in the Weierstrass semigroup at \(Q\). In [J. Pure Appl. Algebra 162, 273--290 (2001; Zbl 0991.94055)], \textit{M. Homma} and \textit{S. J. Kim} proved similar results for two-point geometric Goppa codes. The present article generalizes these results for arbitrary \(m\)-point codes, that is for codes \(C(D,G)\) where \(G=\mu_1 Q_1+\dots+\mu_m Q_m\). Some interesting supplementary results concerning the structure of Weierstrass semigroups at \(m\) points are also included.
reviewed by: Carlos Munuera
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 14G50
Mathematics Subject Classification ID: 94B65
zbMATH DE Number: 2176204
zbMATH Keywords: geometric Goppa codes
zbMATH Keywords: minimum distance
zbMATH Keywords: Weierstrass semigroups
MaRDI profile type: Publication
full work available at URL: s10623-005-6403-4
OpenAlex ID: W1997825023
cites work: Q3347989
cites work: Weierstrass multiple loci of \(n\)-pointed algebraic curves
cites work: The genus of curves over finite fields with many rational points
cites work: Consecutive Weierstrass gaps and minimum distance of Goppa codes
cites work: Q4027638
cites work: Weierstrass points on certain non-classical curves
cites work: Fermat curves over finite fields
cites work: ALGEBRAICO-GEOMETRIC CODES
cites work: The Weierstrass semigroup of a pair of points on a curve
cites work: Goppa codes with Weierstrass pairs
cites work: On the index of the Weierstrass semigroup of a pair of points on a curve
cites work: The Weierstrass semigroup of a pair of Galois Weierstrass points with prime degree on a curve
cites work: Weierstrass pairs and minimum distance of Goppa codes
cites work: Q4336976
cites work: A note on Hermitian codes over GF(q/sup 2/)
cites work: Q4027642
cites work: On the weight hierarchy of geometric Goppa codes
DOI: 10.1007/S10623-005-6403-4
description: scientific article; zbMATH DE number 2176204