On recognition by spectrum of finite simple linear groups over fields of characteristic 2

A finite group $G$ is said to be recognizable by spectrum, i.e., by the set of element orders, if every finite group $H$ having the same spectrum as $G$ is isomorphic to $G$. We prove that the simple linear groups $L_n(2^k)$ are recognizable by spectrum for $n=2^m\geqslant 32$.