In this paper, let q be an odd prime power. Based on new constacyclic codes which contain their Hermitian duals and Hermitian construction, we construct some classes of quantum MDS codes and quantum codes. When q≡1mod4\documentclass[12pt]{minimal}
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\begin{document}$$q\equiv 1\ \textrm{mod}\ 4$$\end{document}, x and y are a divisor of q-1\documentclass[12pt]{minimal}
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\begin{document}$$q-1$$\end{document} and q+1\documentclass[12pt]{minimal}
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\begin{document}$$q+1$$\end{document}, respectively, we can construct a class of new quantum codes of length n=2xyq2m-1q2-1\documentclass[12pt]{minimal}
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\begin{document}$$n=2xy\frac{q^{2m}-1}{q^2-1}$$\end{document} for odd x,y,m≥3\documentclass[12pt]{minimal}
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\begin{document}$$x,y,m\ge 3$$\end{document}. These codes have larger dimensions than existing codes. In addition, for q with the form 2am±(x2+y2)a-1\documentclass[12pt]{minimal}
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\begin{document}$$2am\pm \sqrt{(x^2+y^2)a-1}$$\end{document} and odd x, y, a with gcd(x,y)=1\documentclass[12pt]{minimal}
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\begin{document}$$gcd(x,y)=1$$\end{document}, we get some quantum MDS codes of length n=q2+1a\documentclass[12pt]{minimal}
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\begin{document}$$n=\frac{q^2+1}{a}$$\end{document}.
