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 $I(f(X^n);Y^n)\le (n-1)\cdot \left(1-h(\alpha )\right)$ for any such $f$. Thus, in this case, the optimal function is of the form $f(x^n)=(x_1,\dots ,x_{n-1})$.

5 pages, accepted ISIT 2017