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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 maximalmutual information between $Y^n$ and any Boolean function $\mathrm{b}(X^n)$ isattained 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 $n-1$ bitquantizer, 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$f(x^n)=(x_1,\ldots,x_{n-1})$.

How to Quantize $n$ Outputs of a Binary Symmetric Channel to $n-1$ Bits?