The generator polynomial of this code is specified in terms of its roots from the Galois field GF(2m) [Lin+83].1 Let α be a primitive element in GF(2m). The generator polynomial g(x) of the t- error correcting BCH code of length 2m – 1 is the lowest-degree polynomial over GF(2), which has as its roots:
Lin, Shu, and Daniel J. Costello, Jr., Error Control Coding Fundamentals and Applications. New York: Prentice-Hall, 1983.
Thus, g(x) has α, α2, α3,…, α2t and their conjugates as all its roots. If φi(x) is the minimal polynomial of αi, then g(x) is the least common multiple of α(x), ...