Title: How to Quantize $n$ Outputs of a Binary Symmetric Channel to $n-1$ Bits?,
Abstract: Suppose that $Y^n$ is obtained by observing a uniform Bernoulli random vector
$X^n$ through a binary symmetric channel with crossover probability $\alpha$.
The "most informative Boolean function" conjecture postulates that the maximal
mutual information between $Y^n$ and any Boolean function $\mathrm{b}(X^n)$ is
attained by a dictator function. In this paper, we consider the "complementary"
case in which the Boolean function is replaced by
$f:\left\{0,1\right\}^n\to\left\{0,1\right\}^{n-1}$, namely, an $n-1$ bit
quantizer, and show that $I(f(X^n);Y^n)\leq (n-1)\cdot\left(1-h(\alpha)\right)$
for any such $f$. Thus, in this case, the optimal function is of the form
$f(x^n)=(x_1,\ldots,x_{n-1})$.