In this paper, we study the Hermitian hulls of generalized Reed-Solomon (GRS) codes over finite fields. For a given class of GRS codes, by extending the length, increasing the dimension, and extending the length and increasing the dimension at the same time, we obtain three classes of GRS codes with Hermitian hulls of arbitrary dimensions. Furthermore, based on some known $q^2$-ary Hermitian self-orthogonal GRS codes with dimension $q-1$, we obtain several classes of $q^2$-ary maximum distance separable (MDS) codes with Hermitian hulls of arbitrary dimensions. It is worth noting that the dimension of these MDS codes can be taken from $q$ to $\frac{n}{2}$, and the parameters of these MDS codes can be more flexible by propagation rules. As an application, we derive three new propagation rules for MDS entanglement-assisted quantum error correction codes (EAQECCs) constructed from GRS codes. Then, from the presently known GRS codes with Hermitian hulls, we can directly obtain many MDS EAQECCs with more flexible parameters. Finally, we present several new classes of (MDS) EAQECCs with flexible parameters, and the distance of these codes can be taken from $q+1$ to $\frac{n+2}{2}$.

翻译：本文研究了有限域上广义 Reed-Solomon（GRS）码的 Hermitian 壳。对于给定的一类 GRS 码，通过扩展长度、增加维数以及同时扩展长度和增加维数，我们得到了三类具有任意维度 Hermitian 壳的 GRS 码。此外，基于一些已知的、维度为 $q-1$ 的 $q^2$ 元 Hermitian 自正交 GRS 码，我们得到了几类具有任意维度 Hermitian 壳的 $q^2$ 元最大距离可分（MDS）码。值得注意的是，这些 MDS 码的维度可以从 $q$ 取到 $\frac{n}{2}$，并且通过传播规则，这些 MDS 码的参数可以更加灵活。作为一个应用，我们推导了由 GRS 码构造的 MDS 纠缠辅助量子纠错码（EAQECC）的三条新传播规则。然后，从目前已知的具有 Hermitian 壳的 GRS 码出发，我们可以直接获得许多参数更灵活的 MDS EAQECC。最后，我们提出了几类具有灵活参数的新（MDS）EAQECC，这些码的距离可以从 $q+1$ 取到 $\frac{n+2}{2}$。