Entanglement-assisted quantum error-correcting (EAQEC) codes are a generalization of standard stabilizer quantum codes that can be obtained from arbitrary classical linear codes based on the entanglement-assisted stabilizer formalism. In this paper, by using generalized Reed–Solomon (GRS) codes, we construct two classes of entanglement-assisted quantum error-correcting MDS (EAQEC MDS) codes with parameters q2-12a,q2-12a-2d+c+2,d;cq,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left[ \left[ \frac{q^2-1}{2a},\frac{q^2-1}{2a}-2d+c+2,d;c\right] \right] _q, \end{aligned}$$\end{document}where q is an odd prime power of the form q=2am-1>3\documentclass[12pt]{minimal}
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\begin{document}$$q=2am-1>3$$\end{document} with m≥2\documentclass[12pt]{minimal}
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\begin{document}$$m\ge 2$$\end{document}, 1≤c≤2a-1\documentclass[12pt]{minimal}
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\begin{document}$$1\le c\le 2a-1$$\end{document} and cm+2≤d≤(a+⌈c2⌉)m\documentclass[12pt]{minimal}
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\begin{document}$$c m+2\le d\le (a+\lceil \frac{c}{2}\rceil )m$$\end{document}, and q2-12a+1,q2-12a+1-2d+c+2,d;cq,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left[ \left[ \frac{q^2-1}{2a+1},\frac{q^2-1}{2a+1}-2d+c+2,d;c\right] \right] _q, \end{aligned}$$\end{document}where q is a prime power of the form q=(2a+1)m-1\documentclass[12pt]{minimal}
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\begin{document}$$q=(2a+1)m-1$$\end{document}, 1≤c≤2a\documentclass[12pt]{minimal}
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\begin{document}$$1\le c\le 2a$$\end{document} and cm+2≤d≤(a+1+⌊c2⌋)m\documentclass[12pt]{minimal}
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\begin{document}$$c m+2\le d\le (a+1+\lfloor \frac{c}{2}\rfloor )m$$\end{document}. The EAQEC MDS codes constructed have much larger minimum distance than the known quantum MDS codes with the same length, and most of them are new in the sense that the parameters of EAQEC codes are different from all the previously known ones. In particular, some of our EAQEC MDS codes have much larger d than the known ones that are of the same length and consume the same number of ebits.
