Elementary Constructions of Best Known Quantum Codes

TL;DR

This paper demonstrates that many quantum codes, previously constructed through complex methods involving extension rings, can be more simply derived directly from cyclic codes over finite fields, advocating for more elementary approaches.

Contribution

The paper shows that direct constructions from cyclic codes over finite fields are often sufficient, simplifying the process compared to more complex ring-based methods.

Findings

Most quantum codes from extension rings can be obtained directly from cyclic codes.

Direct methods are preferable unless explicit benefits of complex approaches are shown.

Simplifies the construction process of quantum codes.

Abstract

Recently, many good quantum codes over various finite fields $F_q$ have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from cyclic codes or their generalizations over $F_q$. Unless explicit benefits are demonstrated for the indirect approach, we believe that direct and more elementary methods should be preferred.