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

Let (R=mathbb {F}_{2^{m}}+umathbb {F}_{2^{m}}+cdots +u^{k}mathbb {F}_{2^{m}}), where (mathbb {F}_{2^{m}}) is the finite field with (2^{m}) elements, m is a positive integer, and 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. In particular, 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.