Transform domain characterization of cyclic codes overZm

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