Orthogonal Groups in Characteristic 2 Acting on Polytopes of High Rank

We show that for all integers \(m\geqslant 2\), and all integers \(k\geqslant 2\), the orthogonal groups \({{\,\mathrm{O}\,}}^{\pm }(2m,\mathbb {F}_{2^k})\) act on abstract regular polytopes of rank 2m, and the symplectic groups \({{\,\mathrm{Sp}\,}}(2m,\mathbb {F}_{2^k})\) act on abstract regular polytopes of rank \(2m+1\).