Galois hulls of constacyclic codes over affine algebra rings

Abstract

Let $\mathcal{A}$ the affine algebra given by the ring ${\mathbb{F}}_q[X_1,X_2,\dots ,X_{\ell }]/I$, where $I$ is the ideal $⟨t_1(X_1),t_2(X_2),\dots ,t_{\ell }(X_{\ell })⟩$ with each $t_i(X_i)$, $1\le i\le \ell$, 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.