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The role of the finite Fourier transform in the theory of error correcting codes has been explored in a recent text by Richard Blahut. In this work we study how the finite Fourier transform relates to certain polynomial identities involving weight enumerator polynomials of linear codes. These include the generalized MacWilliams identities and theorems originally due to R. Gleason concerning polynomial algebras containing weight enumerator polynomials. The Heisenberg group model of the finitedoi:10.1090/s0002-9947-1985-0766218-x fatcat:qtxm4zzmirgdbg76ksshjoto3u