The essential $2$-dimension of the linear groups

In this paper, we compute the essential $2$-dimension when the defining prime is odd of the general linear groups, the projective general linear groups, the special linear groups when $n$ is odd or $n=2$, as well as the special linear groups and quotients of it (such as the projective special linear groups) in the case case $q\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}4$, $s=v_2(q-1)$, and $\mathrm{\Gamma }=\text{Gal}(k({\zeta }_{2^s})\mathrm{/}k)$ is trivial.