Transform domain characterization of cyclic codes over Z_m

Abstract

Cyclic codes with symbols from a residue class integer ring Z_m are characterized in terms of the discrete Fourier transform (DFT) of codewords defined over an appropriate extension ring of Z_m. It is shown that a cyclic code of length n over Z_m, n relatively prime to m, consists of n-tuples over Z_m 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 Z_m 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 Z_m are characterized in the transform domain for cyclic codes. Necessary and sufficient conditions for the existence of self-dual codes over Z_m are obtained and nonexistence of self-dual codes for certain values of m is proved.