On finite field arithmetic in characteristic $2$

We are interested in extending normal bases of $\mathbf{F}_{\!2^n}/\mathbf{F}_{\!2}$ to bases of $\mathbf{F}_{\!2^{nd}}/\mathbf{F}_{\!2}$ which allow fast arithmetic in $\mathbf{F}_{\!2^{nd}}$. This question has been recently studied by Thomson and Weir in case $d$ is equal to $2$. We construct efficient extended bases in case $d$ is equal to $3$ and $4$. We also give conditions under which Thomson-Weir construction can be combined with ours.