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

Let p be a prime number and \(q=p^m\) $q=p^m$ for some positive integer m. In this paper, we find the possible Hermitian hull dimensions of \(\lambda \) $\lambda$ -constacyclic codes over \(R_e={\mathbb {F}}_{q^2}+u{\mathbb {F}}_{q^2} +u^2{\mathbb {F}}_{q^2}+\cdots +u^{e-1}{\mathbb {F}}_{q^2}\) $R_e={\mathbb{F}}_{q^2}+u{\mathbb{F}}_{q^2}+u^2{\mathbb{F}}_{q^2}+\cdots +u^{e-1}{\mathbb{F}}_{q^2}$ , \(u^e=1\) $u^e=1$ where \({\mathbb {F}}_{q^2}\) ${\mathbb{F}}_{q^2}$ is the finite field of \(q^2\) $q^2$ elements, \(e|(q+1)\) $e|(q+1)$ and \(\lambda =\eta _1\alpha _1+\eta _2\alpha _2+\cdots +\eta _e\alpha _e\) $\lambda ={\eta }_1{\alpha }_1+{\eta }_2{\alpha }_2+\cdots +{\eta }_e{\alpha }_e$ for \(\alpha _l \in {\mathbb {F}}_{q^2}^{*}\) ${\alpha }_l\in {\mathbb{F}}_{q^2}^{\ast }$ of order \(r_l\) $r_l$ such that \(r_l\mid q+1\) $r_l\mid q+1$ (for each \(1\le l \le e\) $1\le l\le e$ ). 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.