An Introduction to Reed–Solomon Codes

Abstract

Reed-Solomon codes were the first widely used codes that can correct multiple errors in transmitted information words. They remain useful in many contexts because they can correct bursts of errors to adjacent bits. They do this by turning words made up of bits into sequences of elements of a finite field with q elements where q is large, and working in that finite field. By contrast, Hamming codes work directly with individual bits and are defined over the field of 2 elements. This chapter introduces coding and decoding in a Reed-Solomon code, using Reed and Solomon’s original encoding procedure and the Welch-Berlekamp decoding procedure. The latter reduces to finding solutions of a system of n linear equations in $$n + 1$$n+1unknowns. The theoretical discussion of why decoding works relies on D’Alembert’s Theorem from Chap. 6, and the decoding method requires some elementary knowledge of matrices and solving systems of linear equations, not all of which is found in Sect. 7.1.