Constructions of quantum MDS codes

Let ${\mathbb{F}}_q$ be a finite field with $q=p^e$ elements, where $p$ is a prime number and $e\ge 1$ is an integer. In this paper, by means of generalized Reed-Solomon (GRS) codes, we construct two new classes of quantum maximum-distance-separable ( quantum MDS) codes with parameters $$[[q+1,2k-q-1,q-k+2]{]}_q$$ for $\lceil\frac{q+2}{2}\rceil \leq k\leq q+1$,and$$[[n,2k-n,n-k+1]{]}_q$$for$n\leq q $and$ \lceil\frac{n}{2}\rceil \leq k\leq n$. Our constructions improve and generalize some results of available in the literature. Moreover, we give an affirmative answer to the open problem proposed by Fang et al. in \cite{Fang1}.