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 $${\mathbb {F}}_{2^{2m}}+u{\mathbb {F}}_{2^{2m}}$$.