Some cyclic codes with prime length from cyclotomy of order 4

Cyclic codes are a subclass of linear codes and are widely used in many applications as they have efficient encoding and decoding algorithms. In this paper, we investigate two classes of cyclic codes over 𝔽q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {F}_{q}$\end{document} with prime length using cyclotomy of order 4. Both the dimensions and the generator polynomials are explicitly determined. The method serves as a connection between cyclic codes and sequences over 𝔽q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {F}_{q}$\end{document} whose supports are the unions of certain cyclotomic classes of order 4. The main results thus partially answer two open problems in Ding (2015).