BCH codes; Minimum Polynomials

[Josie12]

Hey,

I need help finding the generator polynomial for a BCH code.

I have been going through a fair few pages over the internet to learn about BCH codes.
Basically the information i have so far is that to get the generator matrix i must consider the roots (nth root of unity).
The generator matrix is the LCM(m(1),m(2),...m(i),...m(2t)) Where m(i) is the minimum polynomial of the root.
All the sites i have looked at have kind of glossed over the minimum polynomials. How do I work these minimum polynomials out?

 

[permute]

example in real/complex:
given 0+i as the element in the extension field (complex), find a minimal polynomial with coefficients in the base field (real). The math will all be done in the extension field (complex) -- only the coefficients of the polynomial as restricted to elements from the base field (real).

One polynomial that has 0+i as a root would be x(x+1)(x^2 +1). This is a 4th degree polynomial. However, it is clear that x(x^2+1), a degree 3 polynomial also has 0+i as a root. Further, (x^2+1) has 0+i as a root. (x - i) is a polynomial of lower order that has 0+i as a root, but it doesn't have all of its coefficients in the base field ( -i is complex). Thus (x^2+1) is a minimal polynomial for 0+i.

This also works for galois fields.

Another related concept is "monic". where (x^2 + 1) and 2x^2 + 2 both have 0+i as a root, "monic" will select the polynomial that has a coefficient of 1 (a concept that exists in any field) as the coefficient of the highest order term.