Quantum codes from cyclic codes over the ring ${\mathbb {F}}_q+v_1{\mathbb {F}}_q+\cdots +v_r{\mathbb {F}}_q$

Abstract

Let $R = {{\mathbb {F}}_q} + {v_1}{{\mathbb {F}}_q} + \cdots + {v_r}{{\mathbb {F}}_q},$ where q is a power of a prime, $v_i^2=v_i,\; v_iv_j=v_jv_i=0$ for $1\le i,j \le r$ and $r\ge 1$. In this paper, the structure of cyclic codes over the ring R is studied and a Gray map $\phi$ from ${R^n}$ to ${\mathbb {F}}_q^{(r + 1)n}$ is given. We give a construction of quantum codes from cyclic codes over the ring R. We derive Euclidean dual containing codes over ${\mathbb {F}}_q$ and Hermitian dual containing codes over ${\mathbb {F}}_{p^{2m}}$ as Gray images of cyclic codes over R. In particular, we use $r+1$ codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code. Furthermore, some new non-binary quantum codes are obtained.