Interest in the hulls of linear codes has been growing rapidly. More is known when the inner product is Euclidean than Hermitian. A shift to the latter is gaining traction. The focus is on a code whose Hermitian hull dimension and dual distance can be systematically determined. Such a code can serve as an ingredient in designing the parameters of entanglement-assisted quantum error-correcting codes (EAQECCs). We use tools from algebraic function fields of one variable to efficiently determine a good lower bound on the Hermitian hull dimensions of generalized rational algebraic geometry (AG) codes. We identify families of AG codes whose hull dimensions can be well estimated by a lower bound. Given such a code, the idea is to select a set of evaluation points for which the residues of the Weil differential associated with the Hermitian dual code has an easily verifiable property. The approach allows us to construct codes with designed Hermitian hull dimensions based on known results on Reed-Solomon codes and their generalization. Using the Hermitian method on these maximum distance separable (MDS) codes with designed hull dimensions yields two families of MDS EAQECCs. We confirm that the excellent parameters of the quantum codes from these families are new.

翻译：近年来，对线性码壳的研究兴趣迅速增长。相较于Hermitian内积，关于Euclidean内积的已知结果更为丰富，但研究重心正逐渐向Hermitian内积转移。本文聚焦于一类能系统确定其Hermitian壳维数与对偶距离的码。此类码可作为设计纠缠辅助量子纠错码（EAQECC）参数的一个要素。我们利用单变量代数函数域的工具，有效确定了广义有理代数几何（AG）码Hermitian壳维数的一个良好下界。我们识别出了一系列AG码族，其壳维数可通过该下界得到较好估计。对于给定的此类码，核心思想是选取一组评估点，使得与Hermitian对偶码相关联的Weil微分的留数具有易于验证的性质。该方法使我们能够基于Reed-Solomon码及其推广的已知结果，构造具有预设Hermitian壳维数的码。将Hermitian方法应用于这些具有预设壳维数的最大距离可分（MDS）码，我们得到了两个MDS EAQECC族。我们确认，这些族所产生的量子码的优异参数是全新的。