Constructions of quasi-twisted quantum codes

TL;DR

This paper introduces new methods for constructing quantum error-correcting codes using quasi-twisted codes, including a generalization of the Hermitian Construction, resulting in codes that outperform existing bounds.

Contribution

It provides a necessary and sufficient condition for Hermitian self-orthogonality of QT codes and introduces a new construction method for q-ary quantum codes, expanding the toolkit for quantum code design.

Findings

Constructed quantum codes exceeding the Quantum Gilbert-Varshamov Bound.

Derived binary quantum codes with improved parameters over current records.

Produced ternary and quaternary codes that fill gaps or outperform existing results.

Abstract

In this work, our main objective is to construct quantum codes from quasi-twisted (QT) codes. At first, a necessary and sufficient condition for Hermitian self-orthogonality of QT codes is introduced by virtue of the Chinese Remainder Theorem (CRT). Then we utilize these self-orthogonal QT codes to provide quantum codes via the famous Hermitian Construction. Moreover, we present a new construction method of q-ary quantum codes, which can be viewed as an effective generalization of the Hermitian Construction. General QT codes that are not self-orthogonal are also employed to construct quantum codes. As the computational results, some binary, ternary and quaternary quantum codes are constructed and their parameters are determined, which all exceed the Quantum Gilbert-Varshamov (GV) Bound. In the binary case, a small number of quantum codes are derived with strictly improved parameters compared with the current records. In the ternary and quaternary cases, our codes fill some gaps or have better performances than the current results.