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 $\frac{n+2}{2}$.

翻译：本文研究了有限域上（扩展）广义里德-所罗门（GRS和EGRS）码的Hermitian壳。针对一类给定的（扩展）GRS码，通过增加长度、增加维数以及同时增加长度和维数，我们得到了三类新的具有任意维Hermitian壳的（扩展）GRS码。此外，我们获得了若干新的具有任意维Hermitian壳的$q^2$元极大距离可分（MDS）码，这些MDS码的维数可以取从$1$到$\frac{n}{2}$的值。通过传播规则，所得码的参数更加灵活。作为应用，可以从已知的（扩展）GRS码构造大量新的（MDS）纠缠辅助量子纠错码（EAQECC）。我们推导了基于（扩展）GRS码构造的（MDS）EAQECC的三条新传播规则。最后，我们提出了若干参数灵活的新型（MDS）EAQECC类。值得注意的是，我们码的距离参数范围可以从$2$到$\frac{n+2}{2}$。