Transform domain characterization of abelian codes

Abstract

Abelian codes constitute a class of codes that includes cyclic codes as a special case. It is shown that the general class of abelian codes can be characterized in the transform domain using the discrete Fourier transform (DFT) over finite fields with the appropriate mixed radix number system as the indexing scheme for DFT coefficients. A simple transform domain description for dual codes of abelian codes is also obtained. Using this description the idempotent generator of the dual of a given abelian code can be easily obtained. Finally, it is shown that in the case of cyclic codes which can be considered as abelian codes also, one can work in smaller extension fields compared to the extension fields if they were considered as cyclic codes only.