From primary to dual affine variety codes over the Klein quartic

TL;DR

This paper extends a method for estimating minimum distances of affine variety codes to dual codes over the Klein quartic, leading to improved parameters for asymmetric quantum codes.

Contribution

It introduces a new approach to analyze dual affine variety codes, providing more precise information and enabling the construction of better asymmetric quantum codes.

Findings

More accurate minimum distance estimates for dual codes

Construction of asymmetric quantum codes with improved parameters

Enhanced understanding of primary and dual affine variety codes

Abstract

In [17] a novel method was established to estimate the minimum distance of primary affine variety codes and a thorough treatment of the Klein quartic led to the discovery of a family of primary codes with good parameters, the duals of which were originally treated in [23][Ex. 3.2, Ex. 4.1]. In the present work we translate the method from [17] into a method for also dealing with dual codes and we demonstrate that for the considered family of dual affine variety codes from the Klein quartic our method produces much more accurate information than what was found in [23]. Combining then our knowledge on both primary and dual codes we determine asymmetric quantum codes with desirable parameters.