**1. Introduction**

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)).

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 methodology for finite geometry codes.

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

link, fiber optic cable, etc). Rather than the message vector *μ*, it is the code vector *C* that is transmitted over a channel where the receiver is only able to observe a received vector *C*ˆ. Ideally, in the absence of any noise, it should be the case that *C*ˆ = *C*. On the other hand, if noise is present on the channel, the method used to transform (i.e. 'encode') the message *μ* into the code vector *C* provides a way to recover *μ* from *C*ˆ. The basic strategy behind ECC is,

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a. Embed a *k* dimensional message vector *μ* in a larger vector space of dimension *n* to create

c. If the channel noise does not cause *C*ˆ to be confused with other possible encodings, the original code vector *C* can be recovered using some predetermined decoding scheme. Conceptually speaking, the *C*ˆ that lies within a predefined noise 'sphere' with respect to the original *C* will be decoded as the (ideally) unique *C*; hence, *μ* can be recovered as well. The size of the noise sphere (which is designed as part of the code) determines how many

The general idea behind ECC then is to find a *<sup>C</sup>* that minimizes ||*<sup>C</sup>* <sup>−</sup> *<sup>C</sup>*ˆ|| ; however, numerically determining the minimum distance solution is wrought with dimensionality issues that can lead to computational intractability. Hence, classes of codes have been devised that relate the message encoding method to the decoding algorithm. Such algorithms are often iterative (Blahut (2003); Lin & Costello (1983); Wicker & Kim (2003)) and converge upon the

Two important quantities in the field of ECC are the Hamming weight and the Hamming

**Definition 3.1.** *The Hamming weight wH*(*v*) *of a vector v is defined as the number of non-zero*

**Definition 3.2.** *The Hamming distance between v and w is defined as the number of components that*

For example, over *GF*(3), assuming *n* = 5, *v* = {02102} and *w* = {02212}, according to

An important quantity for defining the noise sphere is referred to as *dmin* which is the minimum Hamming distance between all code vectors defined in the code class. To correct up to *t* errors in any code vector, it turns out that *dmin* = 2*t* + 1. Furthermore, when the ECC is a linear code, a major simplification arises where *dmin* is simply the minimum Hamming

The GFFT and the convolution theorem have been applied in the field of error control coding for the construction of a class of linear codes known as Reed-Solomon codes (Blahut (2003); Wicker (1994)). The algorithm for encoding a message vector *μ* over *GF*(*pm*) of length *k* is

the above definitions we have that *wH*(*v*) = 3, *wH*(*w*) = 4 and *dH*(*v*, *w*) = 2.

weight computed over all non-zero code vectors in the code class.

**3.1 Application of the GFFT to Reed-Solomon codes**

optimal solution by exploiting the mathematical structure designed into the code.

distance. Consider two vectors *v* and *w* of length *n* over *GF*(*p*).

b. The addition of channel noise converts *C* into the received vector *C*ˆ.

given a message,

*components in v.*

*differ between v and w.*

the code vector *C*.

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

errors can be corrected.

finite geometries. Finally, Sections 5 and 6 conclude this chapter by deriving and applying the generalized convolution theorem.
