Let $\mathcal A$ the affine algebra given by the ring $\mathbb{F}_q[X_1,X_2,\ldots,X_\ell]/ I$, where $I$ is the ideal $\langle t_1(X_1), t_2(X_2), \ldots, t_\ell(X_\ell) \rangle$ with each $t_i(X_i)$, $1\leq i\leq \ell$, being a square-free polynomial over $\mathbb{F}_q$. This paper studies the $k$-Galois hulls of $\lambda$-constacyclic codes over $\mathcal A$ regarding their idempotent generators. For this, first, we define the $k$-Galois inner product over $\mathcal A$ and find the form of the generators of the $k$-Galois dual and the $k$-Galois hull of a $\lambda$-constacyclic code over $\mathcal A$. Then, we derive a formula for the $k$-Galois hull dimension of a $\lambda$-constacyclic code. Further, we provide a condition for a $\lambda$-constacyclic code to be $k$-Galois LCD. Finally, we give some examples of the use of these codes in constructing entanglement-assisted quantum error-correcting codes.

翻译：令 $\mathcal A$ 为仿射代数环 $\mathbb{F}_q[X_1,X_2,\ldots,X_\ell]/ I$，其中 $I$ 是由 $\langle t_1(X_1), t_2(X_2), \ldots, t_\ell(X_\ell) \rangle$ 定义的理想，且每个 $t_i(X_i)$（$1\leq i\leq \ell$）是 $\mathbb{F}_q$ 上的无平方多项式。本文研究了 $\mathcal A$ 上 $\lambda$-常循环码的 $k$-Galois 壳及其幂等生成元。为此，我们首先在 $\mathcal A$ 上定义了 $k$-Galois 内积，并给出了 $\mathcal A$ 上 $\lambda$-常循环码的 $k$-Galois 对偶及其 $k$-Galois 壳的生成元形式。随后，我们推导了 $\lambda$-常循环码的 $k$-Galois 壳维数公式。进一步，我们给出了 $\lambda$-常循环码成为 $k$-Galois LCD 码的条件。最后，我们举例说明了这些码在构建纠缠辅助量子纠错码中的应用。