New quantum codes from dual-containing cyclic codes over finite rings

Abstract

Let $R={\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}+\cdots +u^k{\mathbb{F}}_{2^m}$ , where ${\mathbb{F}}_{2^m}$ is a finite field with $2^m$ elements, $m$ is a positive integer, $u$ is an indeterminate with $u^{k+1}=0.$ In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over $R$. A new Gray map over $R$ is defined and a sufficient and necessary condition for the existence of dual-containing cyclic codes over $R$ is given. A new family of $2^m$-ary quantum codes is obtained via the Gray map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over $R.$ Furthermore, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over $R.$