New quantum codes from constacyclic and additive constacyclic codes

Abstract

Let p be a prime and $q=p^r$, for an integer $r\ge 1$. This article studies $\lambda =(\lambda _1+u\lambda _2+v\lambda _3)$-constacyclic codes of length n over a class of finite commutative non-chain rings $R={\mathbb {F}}_q[u,v]/\langle u^2-\gamma u,v^2-\delta v,uv=vu=0\rangle$, where $\gamma ,\delta \in {\mathbb {F}}_q^{*}$. First, we decompose $(\lambda _1+u\lambda _2+v\lambda _3)$-constacyclic code into the direct sum of $\lambda _1$-constacyclic, $(\lambda _1+\gamma \lambda _2)$-constacyclic and $(\lambda _1+\delta \lambda _3)$-constacyclic codes over ${\mathbb {F}}_q$, respectively. Then, we determine the necessary and sufficient condition for these codes to contain their Euclidean duals. Further, we extend the study to ${\mathbb {F}}_qR$-additive $\lambda$-constacyclic codes of length (n, m) which are R[x]-submodules of $S_{n,m}={\mathbb {F}}_q[x]/\langle x^n-1\rangle \times R[x]/\langle x^m-\lambda \rangle$. Apart from other results, we also discuss the dual-containing separable ${\mathbb {F}}_qR$-additive $\lambda$-constacyclic codes. Finally, by using the CSS construction on the Gray images of these codes, we obtain many new and better quantum codes that improve on the known existing quantum codes available in recent articles.