We study the complexity of computing k-wise independent and
\epsilon-biased generators G : \{0,1\}^n \to \{0,1\}^m.
Specifically, we refer to the complexity of computing G \emph{explicitly}, i.e.
given x \in \{0,1\}^n and i \in \{0,1\}^{\log m}, computing the i-th output bit of G(x).
Mansour, Nisan and Tiwari (1990) show that constant depth circuits of size poly(n) (i.e. AC^0) cannot explicitly compute
k-wise independent and \epsilon-biased generators with seed length n = 2^{\log^{o(1)} m}.

In this work we show that DLOGTIME-uniform constant depth circuits
of size poly(n) \emph{with parity gates} can explicitly compute
k-wise independent and \eps-biased generators with seed length n = (\poly) \log m.
In some cases the seed length of our generators is optimal up to constant factors, and in general up to polynomial factors.
To obtain our results, we show a new construction of combinatorial designs, and we also show how to compute, in DLOGTIME-uniform AC^0, random walks
of length log^c n over certain expander graphs of size 2^n.