Transform domain characterization of Abelian codes

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 discrete Fourier transform (DFT) over finite fields with appropriate mixed radix Manuscript received July 31, 1991. B. S. Rajan is with the Electrical Engineering Department, Indian Institute of Technology, Delhi, New Delhi-110016, India. M. U. Siddiqi is with the Electrical Engineering Department, Indian Institute of Technology, Kanpur-208016, India. IEEE Log Number 9201541. 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.