Application of Polynomials in Coding of Digital Data

Abstract

Communication of information is nowadays ubiquitous in the form of words, pictures and sound that require digital conversion into binary coded signals for transmission due to the requirement of electronic circuitry of the implementing devices and machines. In a subtle way, polynomials play a very deep, important role in generating such codes without errors and distortion over long multiple pathways. This chapter surveys the very basics of such polynomials-based coding for easy grasp in the realization of such a complicated technologically important task. The coding is done in terms of just 0 and 1 that entails use of the algebra of finite Galois fields, using polynomial rings in particular. Even though more than six decades have passed since the foundations of the subject of the theory of coding were laid, the interest in the subject has not diminished as is apparent from the continued appearance of books on the subject. Beginning with the introduction of the ASCII codification of alpha-numeric and other symbols developed for early generation computers, this article proceeds with the application of algebraic methods of coding of linear block codes and the polynomials-based cyclic coding, leading to the development of the BCH and the Reed–Solomon codes, widely used in practices.