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.

Keywords

Galois fields Finite field extension Conjugate classes Minimal polynomials Order

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