Let C be a cyclic (n,k) code with generator g.

If d is a polynomial and a element of S = R[X]/(d), then g(a) has a meaning in S. In particular, if d = g is irreducible, and a = X + (g), then g(a) = 0 in S. So, S has become a field containing a root of g.

Fact (BCH bound)

Let g be an irreducible divisor of Xn - 1 in Z/2Z[X]. Let a = X + (g) be an element of S = Z/2Z[X]/(g).

Put    J = {j is elt of N | g(aj) = 0}.

If J has a sequence of c consecutive integers, then the minimal distance of the code generated by g is at least c + 1.


By choosing the generating polynomial in a clever way, codes can be constructed that correct multiple errors.

BCH stands for Bose, Ray-Chaudhuri and Hocquenghem, the three mathematicians who discovered the bound.