I graded Homework #3. The mean and median were both 39/50, with a high score of 48/50.

Kudos to Runting Shi and Yi Wu for (implicitly, by giving correct answers) noticing that the weighting function in (6b) is a sum of Kravchuk polynomials, not a Kravchuk polynomials as I mistakenly wrote in the solutions.

Kudos also to Ryan Williams for improving the upper bound in problem 1 in some cases: He showed $\mathbb{I}(f) \leq (2 - 2\alpha/(1+\alpha)) w$ when $f$ is a DNF of width $w = \alpha n$. Note that this is tight in the case $\alpha = 1$, by the Parity function.

Open problem note: I'm guessing that indeed $\mathbb{I}(f) \leq w$ for any width-$w$ DNF. Might be a nice thing to try to prove, although I couldn't guarantee that it isn't already known.

## Tuesday, March 20, 2007

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