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

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 (Grassl in bounds on the minimum distance of linear codes and quantum codes. http://​www.​codetables.​de). In nonbinary cases, many quantum codes are new or have better parameters than those in the literature.