On construction of quantum codes with dual-containing quasi-cyclic codes

TL;DR

This paper introduces a new class of dual-containing quasi-cyclic codes and demonstrates their effectiveness in constructing quantum codes with improved parameters over small fields, surpassing existing bounds.

Contribution

It proposes a novel class of 2-generator quasi-cyclic codes with conditions for dual-containment, enabling the construction of quantum codes with better parameters than previous methods.

Findings

Many new quantum codes exceeding the quantum Gilbert-Varshamov bound.

16 binary quantum codes improve the minimum distance lower bound in Grassl's table.

Numerous nonbinary quantum codes with superior parameters are constructed.

Abstract

One of the main objectives of quantum error-correction theory is to construct quantum codes with optimal parameters and properties. In this paper, we propose a class of 2-generator quasi-cyclic codes and study their applications in the construction of quantum codes over small fields. Firstly, some sufficient conditions for these 2-generator quasi-cyclic codes to be dual-containing concerning Hermitian inner product are determined. Then, we utilize these Hermitian dual-containing quasi-cyclic codes to produce quantum codes via the famous Hermitian construction. Moreover, we present a lower bound on the minimum distance of these quasi-cyclic codes, which is helpful to construct quantum codes with larger lengths and dimensions. As the computational results, many new quantum codes that exceed the quantum Gilbert-Varshamov bound are constructed over $F_q$, where $q$ is $2,3,4,5$. In particular, 16 binary quantum codes raise the lower bound on the minimum distance in Grassl's table \cite{Grassl:codetables}. In nonbinary cases, many quantum codes are new or have better parameters than those in the literature.