Galois hulls of constacyclic codes over affine algebra rings
Indibar Debnath, Habibul Islam, Edgar Martínez-Moro, Om Prakash
TL;DR
The paper advances the theory of Galois hulls by studying $k$-Galois hulls of $\\lambda$-constacyclic codes over the semisimple affine algebra $\\mathcal{A}$, exploiting a primitive idempotent decomposition to reduce problems to component codes over $\\mathbb{F}_q$. It defines the $k$-Galois inner product on $\\mathcal{A}^n$, derives explicit generators for $C^{\\perp_k}$ and $\\mathrm{hull}_k(C)$ in terms of component polynomials, and provides a formula for the hull’s $q$-dimension. The work also gives LCD and self-duality criteria, rooted in the factorization of $x^n-\\lambda$ and componentwise analysis, and demonstrates applications to entanglement-assisted quantum error-correcting codes via a Gray map that preserves hull structure. Collectively, these results enable construction and analysis of EAQECCs with controlled hull dimensions, including explicit examples yielding MDS and Near-MDS codes, thereby linking ring-based code alphabets to quantum information tasks.
Abstract
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 $λ$-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 $λ$-constacyclic code over $\mathcal A$. Then, we derive a formula for the $k$-Galois hull dimension of a $λ$-constacyclic code. Further, we provide a condition for a $λ$-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.