New MDS Entanglement-Assisted Quantum Codes from MDS Hermitian Self-Orthogonal Codes

The intersection ${\bf C}\bigcap {\bf C}^{\perp_H}$ of a linear code ${\bf C} \subset {\bf F}_{q^2}$ and its Hermitian dual ${\bf C}^{\perp_H}$ is called the Hermitian hull of this code. A linear code ${\bf C} \subset {\bf F}_{q^2}$ satisfying ${\bf C} \subset {\bf C}^{\perp_H}$ 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 $0 \leq h \leq k$, 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 $d\leq \frac{n+2}{2}$ exist for arbitrary length $n \leq q^2+1$. Moreover any QECC constructed from $k$-dimensional Hermitian self-orthogonal codes can be transformed to $k$ different EAQEC codes.