Which Boolean Functions are Most Informative?

Gowtham R. Kumar; Thomas A. Courtade

Which Boolean Functions are Most Informative?

Information Theory 2013-07-16 v2 math.IT

Authors: Gowtham R. Kumar , Thomas A. Courtade

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 $\alpha$ . For any Boolean function $b:\{0,1\}^n\rightarrow \{0,1\}$ , we conjecture that $I(b(X^n);Y^n)\leq 1-H(\alpha)$ . While the conjecture remains open, we provide substantial evidence supporting its validity.

Keywords

boolean functions

Cite

@article{arxiv.1302.2512,
  title  = {Which Boolean Functions are Most Informative?},
  author = {Gowtham R. Kumar and Thomas A. Courtade},
  journal= {arXiv preprint arXiv:1302.2512},
  year   = {2013}
}

Comments

5 pages, 1 figure. Presented at ISIT 2013 in Istanbul, Turkey. (v2 corrects minor typos present in v1)

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