We develop a new framework for constructing explicit hitting-set generators (HSGs) for classes of Boolean functions. The framework is inspired by the Ajtai–Wigderson approach based on iterated pseudorandom restrictions. We use this framework to construct HSGs for read-once CNFs and read-once CNFs with parities. For formulas on $n$ variables with density at least $\varepsilon$, our generators use $O(\log(n/\varepsilon))$ seed bits. This bound is optimal up to a constant factor. Applying the hitting reduction of Gopalan, Meka, Reingold, Trevisan, and Vadhan gives the same seed bound for general ordered width-3 read-once branching programs. For this class, our construction is the first to achieve optimal dependence on both length and acceptance density. The result provides further evidence supporting the longstanding conjecture $\mathrm{L} = \mathrm{RL}$

Bypassed an unnecessary epsilon <= 1/2 limitation; Updated ECCC abstract that was truncated previously.

We use this framework to construct HSGs for read-once CNFs and read-once CNFs with parities. For formulas on $n$ variables with density at least $\varepsilon$, our generators use $O(\log(n/\varepsilon))$ seed bits. This bound is optimal up to a constant factor. Applying the hitting reduction of Gopalan, Meka, Reingold, Trevisan, and Vadhan gives the same seed bound for general ordered width-3 read-once branching programs. For this class, our construction is the first to achieve optimal dependence on both length and acceptance density. The result provides further evidence supporting the longstanding conjecture $\mathrm{L} = \mathrm{RL}$