Nonbinary quantum codes from constacyclic codes over polynomial residue rings

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