Bounding the Minimum Distance of Affine Variety Codes Using Symbolic Computations of Footprints
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  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.
KeywordsAffine variety codes Buchberger’s algorithm Klein curve Minimum distance
The authors gratefully acknowledge the support from The Danish Council for Independent Research (Grant No. DFF–4002-00367). They are also grateful to Department of Mathematical Sciences, Aalborg University for supporting a one-month visiting professor position for the second listed author. The research of Ferruh Özbudak has been funded by METU Coordinatorship of Scientific Research Projects via grant for projects BAP-01-01-2016-008 and BAP-07-05-2017-007.
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