Three classes of propagation rules for GRS and EGRS codes and their applications to EAQECCs

In this paper, we study the Hermitian hulls of (extended) generalized Reed-Solomon (GRS and EGRS) codes over finite fields. For a given class of (extended) GRS codes, by increasing the length, increasing the dimensions and increasing both the length and the dimensions, we obtain three new classes of (extended) GRS codes with Hermitian hulls of arbitrary dimensions. Furthermore, we obtain several new classes of $q^2$-ary maximum distance separable (MDS) codes with Hermitian hulls of arbitrary dimensions. And the dimension of these MDS codes can be taken from $1$ to $\frac{n}{2}$. By propagation rules, the parameters of the obtained code can be more flexible. As an application, a lot of new (MDS) entanglement-assisted quantum error correction codes (EAQECCs) can be constructed from previous known (extended) GRS codes. We derive three new propagation rules on (MDS) EAQECCs constructed from (extended) GRS codes. Finally, we present several new classes of (MDS) EAQECCs with flexible parameters. Notably, the distance parameters of our codes can range from $2$ to $\frac{n+2}{2}$.