Several new families of MDS EAQECCs with much larger dimensions and related application to EACQCs

Abstract

In 2006, Brun, Devetak and Hsieh developed the theory of entanglement-assisted quantum error-correcting codes (EAQECCs), which generalizes that of quantum stabilizer codes. The study of EAQECCs is currently a hot topic in coding theory and quantum information theory. Let $q=p^{e}$ be a prime power and let $\ell$ be an integer with $0\le \ell \le e-1$. The $\ell$-Galois hulls of MDS codes are an effective tool for constructing MDS EAQECCs. In this paper, we construct twenty-two families of q-ary MDS codes with $\ell$-Galois hulls satisfying $2(e-\ell )\mid e$, which are not covered by those available in Cao (Galois hulls of MDS codes and their quantum error correction, 2020. https://arxiv.org/abs/2002.12892v2, IEEE Trans Inf Theory 67(12):7964–7984, 2021) when $\ell \ne \frac{e}{2}$. From these families of MDS codes with $\ell$-Galois hulls, we obtain twenty-two families of q-ary MDS EAQECCs. Compared with the MDS EAQECCs from Cao (Galois hulls of MDS codes and their quantum error correction, 2020. https://arxiv.org/abs/2002.12892v2, IEEE Trans Inf Theory 67(12):7964–7984, 2021), some of the MDS EAQECCs constructed in this paper have much larger dimensions. As an application, we give new families of p-ary entanglement-assisted concatenated quantum codes (EACQCs) by combining an arbitrary p-ary $[[n',e,d';c']]_{p}$ EAQECC with the q-ary MDS EAQECCs constructed in this paper. It turns out that the parameters of these EACQCs are very flexible, e.g., the possibility of the lengths of these EACQCs is far greater than the one of the lengths of these MDS EAQECCs.