Hermitian hull of constacyclic codes over a class of non-chain rings and new quantum codes

Abstract

Let p be a prime number and $q=p^m$ for some positive integer m. In this paper, we find the possible Hermitian hull dimensions of $\lambda$-constacyclic codes over $R_e={\mathbb {F}}_{q^2}+u{\mathbb {F}}_{q^2} +u^2{\mathbb {F}}_{q^2}+\cdots +u^{e-1}{\mathbb {F}}_{q^2}$, $u^e=1$ where ${\mathbb {F}}_{q^2}$ is the finite field of $q^2$ elements, $e|(q+1)$ and $\lambda =\eta _1\alpha _1+\eta _2\alpha _2+\cdots +\eta _e\alpha _e$ for $\alpha _l \in {\mathbb {F}}_{q^2}^{*}$ of order $r_l$ such that $r_l\mid q+1$ (for each $1\le l \le e$). Further, we obtain some conditions for these codes to be Hermitian LCD. Also, under certain conditions, we establish a strong result that converts every constacyclic code to a Hermitian LCD code (Corollaries 2 and 3). We also study the structure of generator polynomials for Hermitian dual-containing constacyclic codes (Theorems 8 and 9), and obtain parameters of quantum codes using the Hermitian construction. The approach we used to derive Hermitian dual-containing conditions via the hull has not been used earlier. As an application, we obtain several optimal and near-to-optimal LCD codes, constacyclic codes having small hull dimensions, and quantum codes.