Codes from flag varieties over a finite field
type: Item
instance of: scholarly article
title: Codes from flag varieties over a finite field
zbMATH Open document ID: 1021.94027
DOI: 10.1016/S0022-4049(02)00188-3
published in: Journal of Pure and Applied Algebra
publication date: 2003-03-09T00:00:00Z
review text: Algebraic-geometric error-correcting codes are usually constructed from algebraic curves over finite fields, as they were originally introduced by \textit{V. Goppa} [Math. USSR, Izv. 21, 75--91 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, 762--781 (1982; Zbl 0522.94013)]. \textit{S. Vladut} and \textit{Yu. Manin} [J. Soviet. Math. 30, 2611--2643 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 25, 209--257 (1984; Zbl 0629.94013)] showed how to construct codes from higher-dimensional varieties. In this article the author presents a family of codes arising from flag varieties over finite fields. To that end he follows the usual way of embedding a flag variety \(X\) into a projective space and then evaluating linear forms on \(X\) (as described in the book by \textit{M. Tsfasman} and \textit{S. Vladut} [Algebraic-geometric codes, Math. Appl., Sov. Ser. 58 (Dordrecht, Kluwer) (1991; Zbl 0727.94007)]). He also gives the parameters of the obtained codes in some cases.
reviewed by: Carlos Munuera
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 1878477
zbMATH Keywords: algebraic-geometric codes
zbMATH Keywords: error-correcting codes
zbMATH Keywords: flag varieties
zbMATH Keywords: finite fields
author: François Rodier
MaRDI profile type: Publication
cites work: Q3204429
cites work: Reed-Muller codes on complete intersections.
cites work: Q4485866
cites work: Error-correcting codes from higher-dimensional varieties
cites work: The weight hierarchy of higher dimensional Hermitian codes
cites work: The parameters of projective Reed-Müller codes
cites work: Q4336971
cites work: Linear codes and modular curves
cites work: Q4879977
cites work: Number of points of Deligne-Lusztig surfaces
cites work: Q4302493
cites work: Projective Reed-Muller codes
cites work: Q3997777
full work available at URL: s0022-4049(02)00188-3
OpenAlex ID: W2024108989
description: scientific article; zbMATH DE number 1878477