Galois Fields

Abstract

The algebraic code design is achieved using the Galois field. BCH and Reed-Solomon block codes, which are cyclic linear block codes, are designed in an algebraic manner, and their constructions are based on Galois fields. For this reason, it is very important to fully comprehend the topic of Galois fields before proceeding with the construction of algebraic codes, i.e., the codes designed in an algebraic manner. In this chapter, we first provide information about the finite fields and extension of finite fields, and for this purpose, we give the definitions of irreducible polynomials and primitive polynomials which are used for the construction of extended fields. In sequel, we provide information about conjugate classes employed for the construction of minimal polynomials which are utilized for the determination of the generator polynomials of the BCH and Reed-Solomon codes, and these codes are used in many practical communication and data storage devices.

Problems

• 1. Decide whether the polynomials

$$ {x}^4+x+1\kern1.5em {x}^4+{x}^2+x+1\kern1.5em $$

are irreducible polynomials or not.

• 2. Construct the extended field GF(2⁴) using the primitive polynomial p(x) = x⁴ + x³ + 1. Repeat the construction process using the primitive polynomial p(x) = x⁴ + x + 1. Comment on the field elements for both constructions.

• 3. Decide whether the polynomial

$$ p(x)={x}^5+{x}^4+{x}^2+x+1 $$

is a primitive polynomial or not.

• 4. Find all the conjugate classes of GF(2⁵). How many minimal polynomials do we have in GF(2⁵)?

• 5. Obtain the binary representation of the polynomial

$$ p\left(\alpha \right)={\alpha}^2+\alpha $$

in GF(2³) and GF(2⁵).

• 6. Assume that the extended field GF(2⁵) is constructed using the primitive polynomial p(x) = x⁵ + x³ + 1. Obtain the conjugates of α³ in GF(2⁵), and find the minimal polynomial for the obtained conjugate class.

• 7. Factorize x¹⁵ + 1.

• 8. Expand (x⁴ + α²x³ + α⁵x + α³)⁴ in GF(2³).

• 9. Obtain the binary representation of the polynomial p(x) = x⁴ + α⁴x³ + α¹²x + α⁹ in GF(2³) and GF(2⁵). Assume that the primitive polynomials p(x) = x³ + x + 1 and p(x) = x⁵ + x³ + x² + x + 1 are used for the construction of the extended fields GF(2³) and GF(2⁵).

• 10. Evaluate the inverses of α², α² + 1 in GF(2³). Use the primitive polynomial p(x) = x³ + x² + 1.

• 11. Calculate $\sqrt[3]{\alpha^4},\sqrt[2]{\alpha^3},\sqrt[5]{\alpha^2}$ in GF(2³).

• 12. Find the roots of x³ + α³x² + α⁵x + α = 0 in GF(2⁴). Determine the p(x) used to construct the extended field GF(2⁴) by yourself.

• 13. Using p(x) = x² + x + 1, construct GF(2²). Find all the conjugate classes; calculate the minimal polynomials.

• 14. Solve the equation set

$$ {\alpha}^3x+\alpha y={\alpha}^2 $$

$$ {\alpha}^5x+{\alpha}^2y={\alpha}^4 $$

in GF(2³). Use p(x) = x³ + x + 1 as your primitive polynomial.

• 15. Find the determinant of the matrix

$$ A=\left[\begin{array}{cc}\alpha & {\alpha}^2\\ {}{\alpha}^3& {\alpha}^8\end{array}\right] $$

in GF(2⁴). Use p(x) = x⁴ + x + 1 as your primitive polynomial.

• 16. Calculate the inverse of the matrix

$$ A=\left[\begin{array}{cc}{\alpha}^4& {\alpha}^2\\ {}{\alpha}^5& {\alpha}^8\end{array}\right] $$

in GF(2⁴). Use p(x) = x⁴ + x + 1 as your primitive polynomial.

• 17. Calculate the inverse of the matrix

$$ A=\left[\begin{array}{ccc}{\alpha}^2& {\alpha}^6& {\alpha}^5\\ {}{\alpha}^3& {\alpha}^4& {\alpha}^4\\ {}{\alpha}^7& {\alpha}^9& {\alpha}^{12}\end{array}\right] $$

in GF(2⁴). Use p(x) = x⁴ + x + 1 as your primitive polynomial.

• 18. Find the solution of the equation set

$$ {\alpha}^3x+{\alpha}^9y+\alpha z={\alpha}^5 $$

$$ {\alpha}^7x+{\alpha}^2y+{\alpha}^2z={\alpha}^{13} $$

$$ {\alpha}^3x+{\alpha}^4y+{\alpha}^3z={\alpha}^6 $$

in GF(2⁴). Use p(x) = x⁴ + x + 1 as your primitive polynomial.