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=p(k)n (it prime to p) over the ring Z(p)e for prime divisors p of N. We then use the discrete Fourier transform to obtain an isomorphism gamma between Z(p)e[X]/< X-N - 1 > and a direct surn circle plus Si-i is an element of I of certain local rings which are ambient spaces for codes of length p(k) 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 Z(p)e from the ideals of S-i. The inverse isomorphism of gamma 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 S-i are classified in terms of the sequence of their torsion codes. (c) 2005 Elsevier Inc. All rights reserved.