New MDS Entanglement-Assisted Quantum Codes from MDS Hermitian Self-Orthogonal Codes
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The intersection C⋂ C^⊥_H of a linear code C⊂ F_q^2 and its Hermitian dual C^⊥_H is called the Hermitian hull of this code. A linear code C⊂ F_q^2 satisfying C⊂ C^⊥_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 ≤ h ≤ 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≤n+2/2 exist for arbitrary length n ≤ q^2+1. Moreover any QECC constructed from k-dimensional Hermitian self-orthogonal codes can be transformed to k different EAQEC codes.
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