Evaluation Codes from an Affine Veriety Code Perspective

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