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

Let $$R=\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}+\cdots +u^{k}\mathbb {F}_{2^{m}}$$R=F2m+uF2m+⋯+ukF2m, where $$\mathbb {F}_{2^{m}}$$F2m is the finite field with $$2^{m}$$2m elements, m is a positive integer, and u is an indeterminate with $$u^{k+1}=0.$$uk+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 $$2^{m}$$2m-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.