**2.2 Fourier analysis on** *GF*(*q*)*<sup>n</sup>*

2 Will-be-set-by-IN-TECH

it is clear that two main procedures in the experimental design, that is, the estimation of the effects and the analysis of variance can be executed in a description of experimental design on

This chapter is organized as follows. In Section 2, we give preliminaries that are necessary for this study. In Section 3, we provide an introduction to experimental design and describe the characteristic of the previous model in .experimental design. In Section 4, we propose the new model of experimental design on the basis of an orthonormal system and clarify the

Here, we provide a brief explanation of Fourier analysis on finite Abelian groups. Characters

Let *G* be a finite Abelian group (with additive notation), and let *S*<sup>1</sup> be the unit circle in the complex plane. A character on *<sup>G</sup>* is a complex-valued function <sup>X</sup> : *<sup>G</sup>* <sup>→</sup> *<sup>S</sup>*<sup>1</sup> that satisfies the

Let *Gi*, *i* = 1, 2, . . . , *n*, be Abelian groups of respective orders |*Gi*| = *gi*, *i* = 1, 2, . . . , *n*, *g*<sup>1</sup> ≤

*<sup>i</sup>*=1*Gi and g* =

Since the character group of *G* is isomorphic to *G*, we can index the characters by the elements of *G*, that is, {X*a*(*x*)|*a* ∈ *G*} are the characters of *G*. Note that X**0**(*x*) is the principal character and identically equal to 1. The characters {X*a*(*x*)|*a* ∈ *G*} form an orthonormal system:

*<sup>b</sup>*(*x*) =

Any function *f* : *G* → **C**, where **C** is the field of complex numbers, can be uniquely expressed

*a*∈*<sup>G</sup>*

*<sup>g</sup>* ∑ *x*∈*<sup>G</sup>* *f*(*x*)X <sup>∗</sup>

X*a*(*x*)X <sup>∗</sup>

*f*(*x*) = ∑

*<sup>f</sup><sup>a</sup>* <sup>=</sup> <sup>1</sup>

*n* ∏ *i*=1

1, *a* = *b*,

) ∀*x*, *x* ∈ *G*. (1)

*gi*. (2)

0, *<sup>a</sup>* <sup>=</sup> *<sup>b</sup>*, (3)

*fa*X*a*(*x*), (4)

*a*(*x*) (5)

) = X (*x*)X (*x*

characteristic of the model. Finally, Section 5 concludes this chapter.

X (*x* + *x*

In other words, a character is a homomorphism from *G* to the circle group.

*<sup>G</sup>* <sup>=</sup> <sup>×</sup>*<sup>n</sup>*

1 *<sup>g</sup>* ∑ *x*∈*<sup>G</sup>*

*<sup>b</sup>*(*x*) is the complex conjugate of <sup>X</sup>*b*(*x*).

as a linear combination of the following characters:

the basis of an orthonormal system.

**2.1 Fourier analysis on finite Abelian groups**

are important in the context of finite Fourier series.

**2. Preliminaries**

**2.1.1 Characters**

**2.1.2 Fourier transform**

*g*<sup>2</sup> ≤ ··· ≤ *gn*, and let

condition

where X <sup>∗</sup>

where the complex number

is the *a*-th *Fourier coefficient* of *f* .

Assume that *q* is a prime power. Let *GF*(*q*) be a Galois field of order *q* which contains a finite number of elements. We also use *GF*(*q*)*<sup>n</sup>* to denote the set of all *n*-tuples with entries from *GF*(*q*). The elements of *GF*(*q*)*<sup>n</sup>* are referred to as vectors.

**Example 1.** *Consider GF*(3) = {0, 1, 2}*. Addition and multiplication are defined as follows:*


*Moreover, consider n* = 5*.*

$$GF(\mathfrak{J})^5 = \{00000, 10000, \dots, 22222\},\tag{6}$$

*and* <sup>|</sup>*GF*(3)5<sup>|</sup> <sup>=</sup> <sup>243</sup>*.*

Specifying the group *G* in Section 2.1.2 to be the support group of *GF*(*q*)*<sup>n</sup>* and *g* = *qn*, the relations (3), (4) and (5) also hold over the *GF*(*q*)*<sup>n</sup>* domain.
