Aaronson (STOC 2010) conjectured that almost $k$-wise independence fools constant-depth circuits; he called this the generalised Linial-Nisan conjecture. Aaronson himself later found a counterexample for depth-3 circuits. We give here an improved counterexample for depth-2 circuits (DNFs). This shows, for instance, that Bazzi's celebrated result ($k$-wise independence fools DNFs) cannot be generalised in a natural way. We also propose a way to circumvent our counterexample: We define a new notion of pseudorandomness called local couplings and show that it fools DNFs and even decision lists.
I believe that it is relevant to mention that approximately log-wise independent distributions are NOT necessarily statistically close to being log-wise independent. This fact was proved in the attacked PDF file (dated 2002).
