Editing BCH code (section)

=== General BCH codes ===
General BCH codes differ from primitive narrow-sense BCH codes in two respects.

First, the requirement that <math>\alpha</math> be a primitive element of <math>\mathrm{GF}(q^m)</math> can be relaxed. By relaxing this requirement, the code length changes from <math>q^m - 1</math> to <math>\mathrm{ord}(\alpha),</math> the [[Order (group theory)|order]] of the element <math>\alpha.</math>

Second, the consecutive roots of the generator polynomial may run from <math>\alpha^c,\ldots,\alpha^{c+d-2}</math> instead of <math>\alpha,\ldots,\alpha^{d-1}.</math>

'''Definition.''' Fix a finite field <math>GF(q),</math> where <math>q</math> is a prime power. Choose positive integers <math>m,n,d,c</math> such that <math>2\leq d\leq n,</math> <math>{\rm gcd}(n,q)=1,</math>  and <math>m</math> is the [[multiplicative order]] of <math>q</math> modulo <math>n.</math>

As before, let <math>\alpha</math> be a [[primitive nth root of unity|primitive <math>n</math>th root of unity]] in <math>GF(q^m),</math> and let <math>m_i(x)</math> be the [[minimal polynomial (field theory)|minimal polynomial]] over <math>GF(q)</math> of <math>\alpha^i</math> for all <math>i.</math>
The generator polynomial of the BCH code is defined as the [[least common multiple]] <math>g(x) = {\rm lcm}(m_c(x),\ldots,m_{c+d-2}(x)).</math>

'''Note:''' if <math>n=q^m-1</math> as in the simplified definition, then <math>{\rm gcd}(n,q)</math> is 1, and the order of <math>q</math> modulo <math>n</math> is <math>m.</math>
Therefore, the simplified definition is indeed a special case of the general one.