Introduction to Cryptography with Coding Theory, 3rd Edition

24.9 Reed-Solomon Codes

The Reed-Solomon codes, constructed in 1960, are an example of BCH codes. Because they work well for certain types of errors, they have been used in spacecraft communications and in compact discs.

Let $\mathbf{F}$ be a finite field with $q$ elements and let $n=q-1$. A basic fact from the theory of finite fields is that $\mathbf{F}$ contains a primitive $n$th root of unity $\alpha$. Choose $d$ with $1\le d<n$ and let

$$g(X)=(X-\alpha )(X-{\alpha }^2)\cdots (X-{\alpha }^{d-1})\text{.}$$

This is a polynomial with coefficients in $\mathbf{F}$. It generates a BCH code $C$ over $\mathbf{F}$ of length $n$, called a Reed-Solomon code.

Since $g(\alpha )=\cdots =g({\alpha }^{d-1})=0$, the BCH bound implies that the minimum distance for $C$ is at least $d$. Since $g(X)$ is a polynomial of degree $d-1$, it has at most $d$ nonzero coefficients. Therefore, the codeword corresponding ...