Title: New Quantum codes from constacyclic codes over a general non-chain ring Show Chinese title

Title: 从一般非链环上的常循环码中获得的新量子码

Abstract: Let $q$ be a prime power and let $\mathcal{R}=\mathbb{F}_{q}[u_1,u_2, \cdots, u_k]/\langle f_i(u_i),u_iu_j-u_ju_i\rangle$ be a finite non-chain ring, where $f_i(u_i), 1\leq i \leq k$ 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 $f_i(u_i),$ $1 \leq i \leq k$ 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 $\gcd (n,q)\neq 1.$ Show Chinese abstract

Abstract: 设$q$为一个素数幂，令$\mathcal{R}=\mathbb{F}_{q}[u_1,u_2, \cdots, u_k]/\langle f_i(u_i),u_iu_j-u_ju_i\rangle$为一个有限非链环，其中$f_i(u_i), 1\leq i \leq k$为多项式，不是全部为一次多项式，它们在$\mathbb{F}_{q}$上分解为不同的一次因式。我们对环$\mathcal{R}$上的常循环码进行表征，并研究由此得到的量子码。作为应用，得到了一些新的且优于现有最佳码的量子码。 我们还证明，在从 $\mathcal{R}$上的常循环码构造量子码时，多项式 $f_i(u_i),$ $1 \leq i \leq k$ 的选择是无关的，这仅取决于它们的次数。 显示对于任何 $n$，总是存在量子MDS码 $[[n,n-2,2]]_q$，其中 $\gcd (n,q)\neq 1.$