Towards dimension expanders over finite fields

In this paper we study the problem of explicitly constructing a dimension expander raised by [3]: Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}^n $\end{document} be the n dimensional linear space over the field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}$\end{document}. Find a small (ideally constant) set of linear transformations from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}^n $\end{document} to itself {Ai}i∈I such that for every linear subspace V ⊂ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}^n $\end{document} of dimension dim(V)<n/2 we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dim \left( {\sum\limits_{i \in I} {A_i (V)} } \right) \geqslant (1 + \alpha ) \cdot \dim (V),$\end{document} where α>0 is some constant. In other words, the dimension of the subspace spanned by {Ai(V)}i∈I should be at least (1+α)·dim(V). For fields of characteristic zero Lubotzky and Zelmanov [10] completely solved the problem by exhibiting a set of matrices, of size independent of n, having the dimension expansion property. In this paper we consider the finite field version of the problem and obtain the following results. We give a constant number of matrices that expand the dimension of every subspace of dimension d<n/2 by a factor of (1+1/logn).We give a set of O<(logn) matrices with expanding factor of (1+α), for some constant α>0.