On (1−u ) -cyclic codes over Fpk+uFpkFpk+uFpk

We extend the results of [J.F. Qian, L.N. Zhang, S.X. Zhu, (1+u)(1+u)-constacyclic and cyclic codes over F2+uF2F2+uF2, Appl. Math. Lett. 19 (2006) 820–823. [3]] to codes over the commutative ring R=Fpk+uFpkR=Fpk+uFpk, where pp is prime, k∈Nk∈N and u2=0u2=0. In particular, we prove that the Gray image of a linear (1−u)(1−u)-cyclic code over RR of length nn is a distance-invariant quasicyclic code of index pk−1pk−1 and length pknpkn over FpkFpk. We also prove that if (n,p)=1(n,p)=1, then every code of length pknpkn over FpkFpk which is the Gray image of a linear cyclic code of length nn over RR is permutation-equivalent to a quasicyclic code of index pk−1pk−1.