Asymptotic nonlinearity of vectorial Boolean functions

The nonlinearity of a Boolean function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F: \mathbb{F}_{2}^{m}\rightarrow \mathbb{F}_{2}$\end{document} is the minimum Hamming distance between f and all affine functions. The nonlinearity of a S-box \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f: \mathbb{F}_{2}^{m}\rightarrow \mathbb{F}_{2}^{n}$\end{document} is the minimum nonlinearity of its component (Boolean) functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v\cdot f,\, v\in \mathbb{F}_{2}^{n}\,\backslash \{0\}$\end{document}. This notion quantifies the level of resistance of the S-box to the linear attack. In this paper, the distribution of the nonlinearity of (m, n)-functions is investigated. When n = 1, it is known that asymptotically, almost all m-variable Boolean functions have high nonlinearities. We extend this result to (m, n)-functions.