The Fourier Convolution Theorem over Finite Fields: Extensions of Its Application to Error Control Coding

Linear spectral transform techniques such as the discrete Fourier transform and wavelet analysis over real and complex fields have been routinely applied in the literature (Burrus et al. (1998); Strang & Nuygen (1996)). Furthermore, extensions of these techniques over finite fields (Blahut & Burrus (1991); Caire et al. (1993)) have led to applications in the areas of information theory and error control coding (Blahut (2003); Dodd (2003); Sakk (2002); Wicker (1994)). The goal of this chapter is to review the Galois Field Fourier Transform, the associated convolution theorem and its application in the field of error control coding. In doing so, an interesting connection will be established relating the convolution theorem over finite fields to error control codes designed using finite geometries (Blahut (2003); Lin & Costello (1983); Wicker (1994)).