New MDS entanglement-assisted quantum codes from MDS Hermitian self-orthogonal codes

Abstract

The intersection $\textbf{C}\cap \textbf{C}^{\perp _H}$ of a linear code $\textbf{C} \subset \textbf{F}_{q^2}^n$ and its Hermitian dual $\textbf{C}^{\perp _H}$ is called the Hermitian hull of this code. A linear code $\textbf{C} \subset \textbf{F}_{q^2}^n$ satisfying $\textbf{C} \subset \textbf{C}^{\perp _H}$ is called Hermitian self-orthogonal. Many Hermitian self-orthogonal codes were given for the construction of MDS quantum error correction codes (QECCs). In this paper we prove that for a nonnegative integer h satisfying $0 \le h \le k$, a linear Hermitian self-orthogonal $[n, k]_{q^2}$ code is equivalent to a linear h-dimension Hermitian hull code. Therefore a lot of new MDS entanglement-assisted quantum error correction (EAQEC) codes can be constructed from previous known Hermitian self-orthogonal codes. Actually our method shows that previous constructed quantum MDS codes from Hermitian self-orthogonal codes can be transformed to MDS entanglement-assisted quantum codes with nonzero consumption parameter c directly. We prove that MDS EAQEC $[[n, k, d; c]]_q$ codes with nonzero c parameters and $d\le \frac{n+2}{2}$ exist for arbitrary length n satisfying $n \le q^2+1$. Moreover any QECC constructed from k-dimensional Hermitian self-orthogonal codes can be transformed to k different EAQEC codes. We also prove that MDS entanglement-assisted quantum codes exist for all lengths $n\le q^2+1$.