Nonbinary quantum codes from constacyclic codes over polynomial residue rings

Abstract

Let R be the polynomial residue ring ${\mathbb {F}}_{q^{2}}+u{\mathbb {F}}_{q^{2}}$ , where ${\mathbb {F}}_{q^2}$ 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 ${\mathbb {F}}_{q^{2}}^{p}$ 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 ${\mathbb {F}}_{q^{2}}.$ 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 ${\mathbb {F}}_{2^{2m}}+u{\mathbb {F}}_{2^{2m}}$.