  We give a strongly explicit construction of $\varepsilon$-approximate
$k$-designs for the orthogonal group $\mathrm{O}(N)$ and the unitary group
$\mathrm{U}(N)$, for $N=2^n$. Our designs are of cardinality
$\mathrm{poly}(N^k/\varepsilon)$ (equivalently, they have seed length $O(nk +
\log(1/\varepsilon)))$; up to the polynomial, this matches the number of design
elements used by the construction consisting of completely random matrices.
