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

Abstract

In this paper, all cyclic codes with length p s n, ( n prime to p) over the ring R = F p + uF p +⋯+ u k−1 F p are classified. It is first proved that Tor j ( C) is an ideal of S ¯ = F p m [ ω ] / 〈 ω p s - 1 〉 , so that the structure of ideals over extension ring S u k ( m , ω ) = GR ( u k , m ) [ ω ] / 〈 ω p s - 1 〉 is determined. Then, an isomorphism between R[ X]/〈 X N − 1〉 and a direct sum ⊕ h ∈ I S u k ( m h , ω ) can be obtained using discrete Fourier transform. The generator polynomial representation of the corresponding ideals over F p + uF p +⋯+ u k−1 F p is calculated via the inverse isomorphism. Moreover, torsion codes, MS polynomial and inversion formula are described.