Which Boolean Functions are Most Informative?

Gowtham R. Kumar; Thomas A. Courtade

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arXiv · 1302.2512

Which Boolean Functions are Most Informative?

Gowtham R. Kumar·Thomas A. Courtade

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Abstract

We introduce a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs. Specifically, let $X^n$ be i.i.d. Bernoulli(1/2), and let $Y^n$ be the result of passing $X^n$ through a memoryless binary symmetric channel with crossover probability $α$. For any Boolean function $b:\{0,1\}^n\rightarrow \{0,1\}$, we conjecture that $I(b(X^n);Y^n)\leq 1-H(α)$. While the conjecture remains open, we provide substantial evidence supporting its validity.

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Gowtham R. Kumar, Thomas A. Courtade. 2013-07-15. Which Boolean Functions are Most Informative?. https://arxiv.org/abs/1302.2512

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