Definition and illustration[edit]
Primitive narrow-sense BCH codes[edit]
Given a prime number q and prime power qm with positive integers m and d such that d ≤ qm − 1, a primitive narrow-sense BCH code over the finite field (or Galois field) GF(q) with code length n = qm − 1 and distance at least d is constructed by the following method.
Let α be a primitive element of GF(qm).
For any positive integer i, let mi(x) be the minimal polynomial with coefficients in GF(q) of αi.
The generator polynomial of the BCH code is defined as the least common multiple g(x) = lcm(m1(x),…,md − 1(x)).
It can be seen that g(x) is a polynomial with coefficients in GF(q) and divides xn − 1.
Therefore, the polynomial code defined by g(x) is a cyclic code.