Linear recurring sequences and subfield subcodes of cyclic codes

Abstract

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 $\mathcal{M}_{q^m } (f(x))$ denote the set of all linear recurring sequences over $\mathbb{F}_{q^m }$ with characteristic polynomial f(x) over $\mathbb{F}_{q^m }$. Denote the restriction of $\mathcal{M}_{q^m } (f(x))$ to sequences over $\mathbb{F}_q$ and the set after applying trace function to each sequence in $\mathcal{M}_{q^m } (f(x))$ by $\left. {\mathcal{M}_{q^m } (f(x))} \right|_{\mathbb{F}_q }$ and $Tr\left( {\mathcal{M}_{q^m } \left( {f\left( x \right)} \right)} \right)$, respectively. It is shown that these two sets are both complete sets of linear recurring sequences over $\mathbb{F}_q$ with some characteristic polynomials over $\mathbb{F}_q$. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over $\mathbb{F}_{q^m }$.