9 - BCH, Reed–Solomon, and related codes

Summary

Introduction

In Chapter 7 we gave one useful generalization of the (7, 4) Hamming code of the Introduction: the family of (2^m − 1, 2^m − 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 = 2^m − 1 is given by (see Section 7.4)

where (v₀, v₁, …, v_n−1) is some ordering of the 2^m − 1 nonzero (column) vectors from V_m = 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, …, w_n−1 ∈ V_m, will serve as the parity-check matrix for a two-error-correcting code of length n. Since however, the v_i's are distinct, we may view the correspondence v_i → w_i as a function from V_m 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.