Primitive polynomials and m-sequences over GF(q m)

Procedures for obtaining primitive polynomials and m-sequences over GF(q^m) in terms of primitive polynomials and m-sequences over GF(q) are presented. Using a degree mn primitive polynomial g( x) in GF[g, x], an m-sequence over GF(q^m) can be expressed as a vector m-sequence whose component m-sequences are shifted versions of the m-sequence generated by g(x). The degree-n primitive polynomials in GF[q^m,x] with root αqⁱ , that are factors of g(x) with root α when g(x) is viewed in GF[q^m,x ], are then developed from the m-sequence over GF(q ^m). Expressions for the shifts and corresponding primitive polynomial for the m-sequence generated by the u th decimation of the m-sequence generated by the polynomials are also given. Expressions for the shifts and corresponding primitive polynomial factors of g(x) for different bases expressing GF(q^m) over GF(q) are presented