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 S¯=Fpm[ω]/〈ωps-1〉, so that the structure of ideals over extension ring Suk(m,ω)=GR(uk,m)[ω]/〈ωps-1〉Suk(m,ω)=GR(uk,m)[ω]/〈ωps-1〉 is determined. Then, an isomorphism between R[X]/〈XN − 1〉 and a direct sum ⊕h∈ISuk(mh,ω)⊕h∈ISuk(mh,ω) 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.