RS and Binary BCH Codes

Abstract

This chapter begins presenting the original approach to Reed–Solomon (RS) codes exhibiting the generator matrix of the code. This is followed by an equivalent method that selects codewords “screening” words by the parity-check matrix. The familiar ideas of addition and multiplication of complex numbers are utilized to construct Galois fields using irreducible polynomials. As justified in the chapter, only primitive polynomials are used in coding. The concept of the period of a polynomial is introduced, and it is shown how periods can be found with Galois-type linear feedback shift registers (LFSRs). The polynomial approach to coding RS codes is presented, and a method to mechanize coding using LFSRs is explained. The chapter ends introducing the binary Bose–Chaudhuri–Hocquenghem (BCH) codes as the binary codewords of RS codes and proving they are polynomial codes.