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

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 \(n\le q^2+1\).