Galois hulls of constacyclic codes over affine algebra rings

TL;DR

This paper investigates the structure and properties of Galois hulls of constacyclic codes over affine algebra rings, providing formulas and conditions that facilitate their use in quantum error correction.

Contribution

It introduces the concept of $k$-Galois hulls for constacyclic codes over affine algebra rings and derives explicit formulas and conditions for their duals and hulls.

Findings

Formulation of the $k$-Galois inner product over affine algebra rings.

Explicit expressions for the generators of $k$-Galois dual and hull.

Conditions for a code to be $k$-Galois LCD and applications in quantum codes.

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 $\lambda$-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.