On modular cyclic codes

We study cyclic codes of arbitrary length N over the ring of integers modulo M. We first reduce this to the study of cyclic codes of length N=pkn (n prime to p) over the ring Zpe for prime divisors p of N. We then use the discrete Fourier transform to obtain an isomorphism γ between Zpe[X]/〈XN-1〉 and a direct sum ⊕i∈ISi of certain local rings which are ambient spaces for codes of length pk over certain Galois rings, where I is the complete set of representatives of p-cyclotomic cosets modulo n. Via this isomorphism we may obtain all codes over Zpe from the ideals of Si. The inverse isomorphism of γ is explicitly determined, so that the polynomial representations of the corresponding ideals can be calculated. The general notion of higher torsion codes is defined and the ideals of Si are classified in terms of the sequence of their torsion codes.