We prove a new derandomization of Håstad's switching lemma, showing how to efficiently generate restrictions satisfying the switching lemma for DNF or CNF formulas of size $m$ using only $\widetilde{O}(\log m)$ random bits. Derandomizations of the switching lemma have been useful in many works as a key building-block for constructing objects which are in some way provably-pseudorandom with respect to AC$^0$-circuits.

Here, we use our new derandomization to give an improved analysis of the pseudorandom generator of Trevisan and Xue for AC$^0$-circuits (CCC'13): we show that the generator $\varepsilon$-fools size-$m$, depth-$D$ circuits with $n$-bit inputs using only $\widetilde{O}(\log(m/\varepsilon)^{D} \cdot \log n)$ random bits. In particular, we obtain (modulo the $\log \log$-factors hidden in the $\widetilde{O}$-notation) a dependence on $m/\varepsilon$ which is best-possible with respect to currently-known AC$^0$-circuit lower bounds.

Fixed some typos.

We prove a new derandomization of Håstad's switching lemma, showing how to efficiently generate restrictions satisfying the switching lemma for DNF or CNF formulas of size $m$ using only $\widetilde{O}(\log m)$ random bits. Derandomizations of the switching lemma have been useful in many works as a key building-block for constructing objects which are in some way provably-pseudorandom with respect to AC$^0$-circuits.

Here, we use our new derandomization to give an improved analysis of the pseudorandom generator of Trevisan and Xue for AC$^0$-circuits (CCC'13): we show that the generator $\varepsilon$-fools size-$m$, depth-$D$ circuits with $n$-bit inputs using only $\widetilde{O}(\log(m/\varepsilon)^{D} \cdot \log n)$ random bits. In particular, we obtain (modulo the $\log \log$-factors hidden in the $\widetilde{O}$-notation) a dependence on $m/\varepsilon$ which is best-possible with respect to currently-known AC$^0$-circuit lower bounds.