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 be a finite field with elements and let . A basic fact from the theory of finite fields is that contains a primitive th root of unity . Choose with and let
This is a polynomial with coefficients in . It generates a BCH code over of length , called a Reed-Solomon code.
Since , the BCH bound implies that the minimum distance for is at least . Since is a polynomial of degree , it has at most nonzero coefficients. Therefore, the codeword corresponding ...