A class of binary cyclic codes and sequence families

For two odd integers l, k with 0<l<k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<l<k$$\end{document} and gcd(l,k)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gcd (l,k)=1$$\end{document}, let m=2k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=2k$$\end{document} and d=2lk+12l+1+2(2m-1)3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=\frac{2^{lk}+1}{2^l+1}+\frac{2(2^m-1)}{3}$$\end{document}. In this paper, we determine the value distribution of the exponential sum ∑x∈F2m(-1)Tr1m(ax+bxd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{x\in \mathbb {F}_{2^m}}(-1)^{\mathrm {Tr}_1^m(ax+bx^d)}$$\end{document}. As applications, the weight distribution of a class of binary cyclic codes is settled. Second, we determine the correlation distribution among sequences in a sequence family.