# On affine variety codes from the Klein quartic

## TL;DR

This paper constructs and analyzes a family of affine variety codes from the Klein quartic, showing they have parameters comparable to the best known codes using a novel footprint bound method inspired by Buchberger's algorithm.

## Contribution

Introduces a new method based on symbolic computations and Gr"obner basis theory to determine parameters of affine variety codes from the Klein quartic.

## Key findings

- Most codes have parameters matching the best known codes
- Remaining codes have nearly optimal parameters
- Develops a new footprint bound technique inspired by Buchberger's algorithm

## 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, Ex. 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 [10, 7] from Gr\"obner basis theory and for this purpose we develop a new method where we inspired by Buchberger's algorithm perform a series of symbolic computations. 1