Introduction to Cryptography with Coding Theory, 3rd Edition

24.8 BCH Codes

BCH codes are a class of cyclic codes. They were discovered around 1959 by R. C. Bose and D. K. Ray-Chaudhuri and independently by A. Hocquenghem. One reason they are important is that there exist good decoding algorithms that correct multiple errors (see, for example, [Gallager] or [Wicker]). BCH codes are used in satellites. The special BCH codes called Reed-Solomon codes (see Section 24.9) have numerous applications.

Before describing BCH codes, we need a fact about finite fields. Let $\mathbf{F}$ be a finite field with $q$ elements. From Section 3.11, we know that $q=p^m$ is a power of a prime number $p$. Let $n$ be a positive integer not divisible by $p$. Then it can be proved that there exists a finite field ${\mathbf{F}}^{\prime }$ containing $\mathbf{F}$ such that ${\mathbf{F}}^{\prime }$ contains ...