Two new classes of quantum MDS codes

Let p be a prime and let q be a power of p. In this paper, by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distanceseparable (MDS) codes with parameters [[tq, tq − 2d+ 2, d]]q for any 1 ≤ t ≤ q, 2 ≤ d ≤ ⌊ tq+q−1 q+1 ⌋+ 1, and [[t(q + 1) + 2, t(q + 1)− 2d+ 4, d]]q for any 1 ≤ t ≤ q− 1, 2 ≤ d ≤ t+2 with (p, t, d) 6= (2, q − 1, q). Our quantum codes have flexible parameters, and have minimum distances larger than q 2 +1 when t > q 2 . Furthermore, it turns out that our constructions generalize and improve some previous results.