  In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom
generator for length $n$ and width $w$ read-once branching programs with seed
length $O(\log n\cdot \log(nw)+\log n\cdot\log(1/\varepsilon))$ and error
$\varepsilon$. It remains a central question to reduce the seed length to
$O(\log (nw/\varepsilon))$, which would prove that $\mathbf{BPL}=\mathbf{L}$.
However, there has been no improvement on Nisan's construction for the case
$n=w$, which is most relevant to space-bounded derandomization.
  Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced
the notion of a pseudorandom pseudo-distribution (PRPD) and gave an explicit
construction of a PRPD with seed length $\tilde{O}(\log n\cdot
\log(nw)+\log(1/\varepsilon))$. A PRPD is a relaxation of a pseudorandom
generator, which suffices for derandomizing $\mathbf{BPL}$ and also implies a
hitting set. Unfortunately, their construction is quite involved and
complicated. Hoza and Zuckerman (FOCS'18) later constructed a much simpler
hitting set generator with seed length $O(\log n\cdot
\log(nw)+\log(1/\varepsilon))$, but their techniques are restricted to hitting
sets.
  In this work, we construct a PRPD with seed length $$O(\log n\cdot \log
(nw)\cdot \log\log(nw)+\log(1/\varepsilon)).$$ This improves upon the
construction in [BCG18] by a $O(\log\log(1/\varepsilon))$ factor, and is
optimal in the small error regime. In addition, we believe our construction and
analysis to be simpler than the work of Braverman, Cohen and Garg.
