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Let $$R=\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}+\cdots +u^{k}\mathbb {F}_{2^{m}}$$ , 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. [ABSTRACT FROM AUTHOR]

Let R be the polynomial residue ring F q 2 + u F q 2 , where F q 2 is the finite field with q 2 elements, q is a power of a prime p, and u is an indeterminate with u 2 = 0. We introduce a Gray map from R to F q 2 p and study (1 - u) -constacyclic codes over R. It is proved that the image of a (1 - u) -constacyclic code of length n over R under the Gray map is a distance-invariant linear cyclic code of length pn over F q 2. We give some necessary and sufficient conditions for (1 - u) -constacyclic codes over R to be Hermitian dual-containing. In particular, a new class of 2 m -ary quantum codes is obtained via the Gray map and the Hermitian construction from Hermitian dual-containing (1 - u) -constacyclic codes over the ring F 2 2 m + u F 2 2 m . [ABSTRACT FROM AUTHOR]