Introduction

In Chapter 7 we gave one useful generalization of the (7, 4) Hamming code of the Introduction: the family of (2m − 1, 2m − m − 1) single-error-correcting Hamming codes. In Chapter 8 we gave a further generalization, to a class of codes capable of correcting a single burst of errors. In this chapter, however, we will give a far more important and extensive generalization, the multipleerror-correcting BCH2 and Reed–Solomon codes.

To motivate the general definition, recall that the parity-check matrix of a Hamming code of length n = 2m − 1 is given by (see Section 7.4)

where (v0, v1, …, vn−1) is some ordering of the 2m − 1 nonzero (column) vectors from Vm = GF(2)m. The matrix H has dimensions m × n, which means that it takes m parity-check bits to correct one error. If we wish to correct two errors, it stands to reason that m more parity checks will be required. Thus we might guess that a matrix of the general form

where w0, w1, …, wn−1 ∈ Vm, will serve as the parity-check matrix for a two-error-correcting code of length n. Since however, the vi's are distinct, we may view the correspondence vi → wi as a function from Vm into itself, and write H2 as

But how should the function f be chosen? According to the results of Section 7.3, H2 will define a two-error-correcting code iff the syndromes of the error pattern of weights 0, 1 and 2 are all distinct.