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

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