New Quantum codes from constacyclic codes over a general non-chain ring

Abstract

Let $q$ be a prime power and let $\mathcal{R}={\mathbb{F}}_q[u_1,u_2,\cdots ,u_k]/⟨f_i(u_i),u_i{u}_j-u_j{u}_i⟩$ be a finite non-chain ring, where $f_i(u_i),1\le i\le k$ are polynomials, not all linear, which split into distinct linear factors over ${\mathbb{F}}_q$. We characterize constacyclic codes over the ring $\mathcal{R}$ and study quantum codes from these. As an application, some new and better quantum codes, as compared to the best known codes, are obtained. We also prove that the choice of the polynomials $f_i(u_i),$ $1\le i\le k$ is irrelevant while constructing quantum codes from constacyclic codes over $\mathcal{R}$, it depends only on their degrees. It is shown that there always exists Quantum MDS code $[[n,n-2,2]{]}_q$ for any $n$ with $\gcd (n,q)\ne 1.$