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≡1mod4q-1q+1n=2xyq2m-1q2-1x,y,m≥32am±(x2+y2)a-1gcd(x,y)=1n=q2+1a, x and y are a divisor of q≡1mod4q-1q+1n=2xyq2m-1q2-1x,y,m≥32am±(x2+y2)a-1gcd(x,y)=1n=q2+1a and q≡1mod4q-1q+1n=2xyq2m-1q2-1x,y,m≥32am±(x2+y2)a-1gcd(x,y)=1n=q2+1a, respectively, we can construct a class of new quantum codes of length q≡1mod4q-1q+1n=2xyq2m-1q2-1x,y,m≥32am±(x2+y2)a-1gcd(x,y)=1n=q2+1a for odd q≡1mod4q-1q+1n=2xyq2m-1q2-1x,y,m≥32am±(x2+y2)a-1gcd(x,y)=1n=q2+1a. These codes have larger dimensions than existing codes. In addition, for q with the form q≡1mod4q-1q+1n=2xyq2m-1q2-1x,y,m≥32am±(x2+y2)a-1gcd(x,y)=1n=q2+1a and odd x, y, a with q≡1mod4q-1q+1n=2xyq2m-1q2-1x,y,m≥32am±(x2+y2)a-1gcd(x,y)=1n=q2+1a, we get some quantum MDS codes of length q≡1mod4q-1q+1n=2xyq2m-1q2-1x,y,m≥32am±(x2+y2)a-1gcd(x,y)=1n=q2+1a.