We show examples of total Boolean functions that depend on $n$ variables and have spectral sensitivity $\Theta(\sqrt{ \log n})$, which is asymptotically minimal. Our main new function combines the Hamming code with the Boolean address function and has $\lambda(f) = \sqrt{ (1+o(1)) \log_2 n}$, which is optimal even up to a constant factor. By combining this function with itself in a specific way, we also obtain a family of functions with $\text{ s}_0(f) = (c+o(1)) \log_2 n$ and $\text{ s}_0(f) = (1-c+o(1)) \log_2 n$ for any $c \in [0,1]$. This is an optimal tradeoff for Boolean functions with low sensitivity, as the lower bound on sensitivity by Simon generalizes to \[\text{ s}_0(f)+\text{ s}_1(f)\geq\log_2 n - \log_2 \log_2 n + 2.\] As a corollary, this gives a new example of a function with minimal possible sensitivity (up to a constant factor), $\text{ s}(f) = (\frac{ 1}{ 2}+o(1)) \log_2 n$.
Categories: cs.CC