Pseudorandom generator for finite automata
If $d$ is of the order of $n$ then you can write a constant-width branching program as a finite-state automaton, and logarithmic seed length is not known.

But if $d$ is very small, say a constant, then you can do better and achieve logarithmic seed length -- I think, this is something I thought about years ago but never wrote down.  The trick is to use Nisan's result RL $\subseteq$ SC.  Basically, he shows that if you are given a branching program then you can find a logarithmic-seed distribution that fools it.  His result extends to a small number of branching programs.  So if $d$ is a constant then you can enumerate over all possible finite-state automata, and find a distribution that fools all of them.  This should still work as long as the number of programs is polynomial in $n$.