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

Abstract

The intersection $\mathbf{C}\bigcap {\mathbf{C}}^{{\perp }_H}$ of a linear code $\mathbf{C}\subset {\mathbf{F}}_{q^2}$ and its Hermitian dual ${\mathbf{C}}^{{\perp }_H}$ is called the Hermitian hull of this code. A linear code $\mathbf{C}\subset {\mathbf{F}}_{q^2}$ satisfying $\mathbf{C}\subset {\mathbf{C}}^{{\perp }_H}$ 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 $n\le q^2+1$. Moreover any QECC constructed from $k$-dimensional Hermitian self-orthogonal codes can be transformed to $k$ different EAQEC codes.

18 pages, MDS quantum codes can be transformed to MDS Entanglement-assisted quantum codes with nonzero c parameters directly