Cyclic codes over R = Fp + uFp +⋯+ uk−1Fp with length psn

In this paper, all cyclic codes with length psn, (n prime to p) over the ring R = Fp + uFp +⋯+ uk−1Fp are classified. It is first proved that Torj(C) is an ideal of image, so that the structure of ideals over extension ring image is determined. Then, an isomorphism between R[X]/〈XN − 1〉 and a direct sum image can be obtained using discrete Fourier transform. The generator polynomial representation of the corresponding ideals over Fp + uFp +⋯+ uk−1Fp is calculated via the inverse isomorphism. Moreover, torsion codes, MS polynomial and inversion formula are described.