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

Let R = F2m +uF2m + · · ·+u k F2m , where F2m is the finite field with 2 m elements, m is a positive integer, and u is an indeterminate with u = 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-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.