Bounding the minimum distance of affine variety codes using symbolic computations of footprints

Abstract

We study a family of primary affine variety codes defined from the Klein quartic. The duals of these codes have previously been treated in [12, Example 3.2]. Among the codes that we construct almost all have parameters as good as the best known codes according to [9] and in the remaining few cases the parameters are almost as good. To establish the code parameters we apply the footprint bound [7, 10] from Gröbner basis theory and for this purpose we develop a new method where we inspired by Buchbergers algorithm perform a series of symbolic computations.