A Family of Constacyclic Codes over 𝔽 2 m + u 𝔽 2 m and Its Application to Quantum Codes

Abstract

Let R be the ring 𝔽2 m + u𝔽2 m, where u 2 = 0. We introduce a Gray map from R to 𝔽2 2 m and study (1 + u)-constacyclic codes over R. It is proved that the image of a (1 + u)-constacyclic code length n over R under the Gray map is a distance-invariant binary quasicyclic code of index m and length 2mn. We also prove that every code of length 2mn which is the Gray image of cyclic codes over R 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 R.