Abstract
Evaluation codes (also called order domain codes) are traditionally introduced as generalized one-point geometric Goppa codes. In the present paper we will give a new point of view on evaluation codes by introducing them instead as particular nice examples of affine variety codes. Our study includes a reformulation of the usual methods to estimate the minimum distances of evaluation codes into the setting of affine variety codes. Finally we describe the connection to the theory of one-pointgeometric Goppa codes.
Contents
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.2 Affine variety codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.3 Some Gr¨obner basis theoretical tools . . . . . . . . . . . . . . . . . . . . . . . 155
4.4 A bound on the minimum distance of C(I,L) . . . . . . . . . . . . . . . . . . 157
4.5 The Feng-Rao bound for C(I,L)? . . . . . . . . . . . . . . . . . . . . . . . . 160
4.6 Using weighted degree orderings . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.7 The order domain conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.8 Weight functions and order domains . . . . . . . . . . . . . . . . . . . . . . . 171
4.9 Codes form order domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.10 One-point geometric Goppa codes . . . . . . . . . . . . . . . . . . . . . . . . 176
4.11 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
| Originalsprog | Engelsk |
|---|---|
| Titel | Advances In Algebraic Geometry Codes |
| Redaktører | Edgar Martínez Moro, Carlos Munuera Gómez, Diego Ruano Benito |
| Antal sider | 29 |
| Forlag | World Scientific |
| Publikationsdato | 2008 |
| Sider | 153-181 |
| ISBN (Trykt) | 978-9812794000, 981279400X |
| Status | Udgivet - 2008 |
| Navn | Coding Theory and Cryptology |
|---|---|
| Vol/bind | 5 |