3.10 BCH Codes
For the estimation of the minimum distance the following description of the cyclic codes is very useful. Let g(x) be the generator polynomial of the cyclic (n, k) code V over the field 
. As before we assume that the code is nontrivial, that is, 2 
d n and, therefore, n is the period of the polynomial g(x), that is, g(x)|xn – 1 and g(x) does not divide xn’ – 1 for n’ n. Let the code length n and the characteristic p of field be relatively prime. Then the polynomial xn – 1 has no repeated roots since 
. Hence the polynomial g(x) also has no repeated roots because it is the divisor of the polynomial xn – 1. Let m be the minimal positive integer such that n divides qm – 1. Then Fqm is the minimal field containing all roots of g(x). The following theorem called BCH bound holds truth.
Theorem 3.13 Let α be an element of the field Fqm and let αa, αa + 1, …, αa + s – 1 be the roots ...
