Transform domain characterization of cyclic codes over Zm

Cyclic codes with symbols from a residue class integer ring Zm are characterized in terms of the discrete Fourier transform (DFT) of codewords defined over an appropriate extension ring of Zm. It is shown that a cyclic code of length n over Zm, n relatively prime to m, consists of n-tuples over Zm having a specified set of DFT coefficients from the elements of an ideal of a subring of the extension ring. When m is equal to a product of distinct primes every cyclic code over Zm has an idempotent generator and it is shown that the idempotent generators can be easily identified in the transform domain. The dual code pairs over Zm are characterized in the transform domain for cyclic codes. Necessary and sufficient conditions for the existence of self-dual codes over Zm are obtained and nonexistence of self-dual codes for certain values of m is proved.