A family of constacyclic codes over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ and its application to quantum codes

We introduce a Gray map from ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ to ${\mathbb{F}}_2^{2m}$ and study $(1+u)$-constacyclic codes over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m},$ where $u^2=0.$ It is proved that the image of a $(1+u)$-constacyclic code length $n$ over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ under the Gray map is a distance-invariant quasi-cyclic code of index $m$ and length $2mn$ over ${\mathbb{F}}_2\mathrm{.}$ We also prove that every code of length $2mn$ which is the Gray image of cyclic codes over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ of length $n$ is permutation equivalent to a binary quasi-cyclic code of index $m\mathrm{.}$ Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to $(1+u)$-constacyclic codes over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}\mathrm{.}$