A class of binary cyclic codes with five weights

In this paper, the dual code of the binary cyclic code of length 2n − 1 with three zeros α, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha ^{t_1 } $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha ^{t_2 } $$\end{document} is proven to have five nonzero Hamming weights in the case that n ⩾ 4 is even and t1 = 2n/2 + 1, t2 = 2n−1 − 2n/2−1 + 1 or 2n/2 + 3, where α is a primitive element of the finite field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{F}_{2^n } $$\end{document}. The dual code is a divisible code of level n/2 −1, and its weight distribution is also completely determined. When n = 4, the dual code satisfies Ward’s bound.