Linear recurring sequences and subfield subcodes

Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let M_q^m(f(x)) denote the set of all linear recurring sequences over F_q^m with characteristic polynomial f(x) over F_q^m. Denote the restriction of M_q^m(f(x)) to sequences over F_q and the set after applying trace function to each sequence in M_q^m(f(x)) by M_q^m(f(x))|F_q and Tr(M_q^m(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over F_q with some characteristic polynomials over F_q. In this paper, we firstly determine these characteristic polynomials for the two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over F_q^m.