On the structure of order domains

DOI10.1006/ffta.2001.0347zbMath1008.13007OpenAlexW2133164811MaRDI QIDQ700167

Olav Geil, Pellikaan, Ruud

Publication date: 30 September 2002

Published in: Finite Fields and their Applications (Search for Journal in Brave)

Full work available at URL: https://semanticscholar.org/paper/a0ca5f35a87a6e4d4ef0372bc65e7f69e38061eb

zbMATH Keywords

Gröbner basisorder functionorder domainorder structureone-point Goppa codes

Mathematics Subject Classification ID

General structure theory for semigroups (20M10) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) (13P10) Geometric methods (including applications of algebraic geometry) applied to coding theory (94B27) Valuations and their generalizations for commutative rings (13A18) Applications to coding theory and cryptography of arithmetic geometry (14G50)

Related Items (29)

On semigroups, Gröbner basis and algebras admitting a complete set of near weights ⋮ On the evaluation codes given by simple \(\delta \)-sequences ⋮ Evaluation codes and plane valuations ⋮ Near weights on higher dimensional varieties ⋮ On puncturing of codes from norm-trace curves ⋮ Computing the Feng-Rao distances for codes from order domains ⋮ Generalization of the Lee-O'Sullivan list decoding for one-point AG codes ⋮ Further improvements on the Feng-Rao bound for dual codes ⋮ On codes from norm-trace curves ⋮ Relative generalized Hamming weights of evaluation codes ⋮ Integral closures and weight functions over finite fields. ⋮ Evaluation codes defined by finite families of plane valuations at infinity ⋮ Computational aspects of retrieving a representation of an algebraic geometry code ⋮ Geometric and Computational Approach to Classical and Quantum Secret Sharing ⋮ Feng-Rao decoding of primary codes ⋮ Generalized Sudan’s List Decoding for Order Domain Codes ⋮ Evaluation codes and their basic parameters ⋮ Generic interpolation polynomial for list decoding ⋮ Reversely well-ordered valuations on rational function fields in two variables ⋮ The structure of algebras admitting well-agreeing near weights ⋮ List decoding algorithm based on voting in Gröbner bases for general one-point AG codes ⋮ Evaluation codes from order domain theory ⋮ Skeleton simplicial evaluation codes ⋮ Generalized minimum distance functions ⋮ Footprint and minimum distance functions ⋮ Improved evaluation codes defined by plane valuations ⋮ Hilbert quasi-polynomial for order domains and application to coding theory ⋮ The dual of an evaluation code ⋮ An improvement of the Feng-Rao bound for primary codes

Cites Work

• Extension of the Berlekamp-Massey algorithm to N dimensions
• The concept of Gröbner algebras
• On the theory of graded structures
• Syzygies of affine toric varieties
• Generators and relations of abelian semigroups and semigroup rings
• Valuation theory
• Decoding geometric Goppa codes using an extra place
• Lifting standard bases in filtered structures
• Lectures on Polytopes
• A simple approach for construction of algebraic-geometric codes from affine plane curves
• On the decoding of algebraic-geometric codes
• Decoding of codes defined by a single point on a curve
• The minimum distance of codes in an array coming from telescopic semigroups
• Algebraic-geometric codes and multidimensional cyclic codes: a unified theory and algorithms for decoding using Grobner bases
• Shift-register synthesis and BCH decoding
• On the existence of order functions
• New codes for the Berlekamp-Massey-Sakata algorithm
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