Explicit orthogonal and unitary designs

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