**Fourier and Helbert Transform Applications**

228 Fourier Transform Applications

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**1. Introduction**

Wicker (1994)).

methodology for finite geometry codes.

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);

**The Fourier Convolution Theorem** 

**Its Application to Error Control Coding** 

**over Finite Fields: Extensions of** 

*Department of Computer Science, Morgan State University,* 

Eric Sakk and Schinnel Small

*Baltimore, MD,* 

*USA* 

**9**

While a complete exposition of the field of error control would be out of context for this chapter, we refer the interested reader to the recent characterizations of Low-Density Parity Check (LDPC) codes (Pusane et al. (2011); Smarandache et al. (2009); Xia & Fu (2008)). Such formulations have led to a resurgence of interest in the design (Kou et al. (2001); O.Vontobel et al. (2005); Tang et al. (2005); Vandendriesscher (2010)) and decoding (Kou et al. (2001); Li et al. (2010); Liu & Pados (2005); Ngatched et al. (2009); Tang et al. (2005); Zhang et al. (2010)) of finite geometry codes. The formulation in this chapter is meant to serve as a guiding principle relating finite geometric properties to algebraic ones. The vehicle we have chosen to demonstrate these relationships is an example from the field of error control. In particular, we show how a generalized Fourier-like convolution theorem can be applied as a decoding

We begin in Section 2 by reviewing the Galois Field Fourier Transform (GFFT) followed by an overview of error control coding in Section 3. In addition, in Section 3.1 it is demonstrated how the GFFT can be applied within the context of error control coding. Section 4 then goes on to generalize these results to linear transformations using Pascal's triangle as an example. The combinatorics of such a transformation naturally lead to the design of codes derivable from
