Autocorrelation values and Linear complexity of generalized cyclotomic sequence of order four, and construction of cyclic codes

Let $n_1$ and $n_2$ be two distinct primes with $\mathrm{g}\mathrm{c}\mathrm{d}(n_1-1,n_2-1)=4$. In this paper, we compute the autocorrelation values of generalized cyclotomic sequence of order $4$. Our results show that this sequence can have very good autocorrelation property. We determine the linear complexity and minimal polynomial of the generalized cyclotomic sequence over $\mathrm{G}\mathrm{F}(q)$ where $q=p^m$ and $p$ is an odd prime. Our results show that this sequence possesses large linear complexity. So, the sequence can be used in many domains such as cryptography and coding theory. We employ this sequence of order $4$ to construct several classes of cyclic codes over $\mathrm{G}\mathrm{F}(q)$ with length $n_1{n}_2$. We also obtain the lower bounds on the minimum distance of these cyclic codes.