Weil Transform and Error Correcting Codes

摘要

A connection between the Weil transform and error-correcting codes is developed. This contrasts with existing efforts that are based on the Fourier transform and have led to new classes of codes well tailored for burst-error correction. A number of analytical questions must be treated in extending these analogies to the Weil case. In particular, bounds on the mean-square norm of the discrepancy between the FFT (Fast Fourier Transform) and the continuous transform should be established. This was done be consideration of a Riemann sum. The application to information theory then results by applying the Landau-Slepian 'approximate dimension theorems' for functions that are only approximately band-limited. Allegations that certain results on 'Fourier Transforms for Abelian Groups' which we announced as accomplishments on this grant and a related Darpa grant effort, were actually part of a well-known body of tutorial material are examined and refuted.