Skip to main content
×
×
Home

Indistinguishable Sceneries on the Boolean Hypercube

  • RENAN GROSS (a1) and URI GRUPEL (a1)
Abstract

We show that the scenery reconstruction problem on the Boolean hypercube is in general impossible. This is done by using locally biased functions, in which every vertex has a constant fraction of neighbours coloured by 1, and locally stable functions, in which every vertex has a constant fraction of neighbours coloured by its own colour. Our methods are constructive, and also give super-polynomial lower bounds on the number of locally biased and locally stable functions. We further show similar results for ℤn and other graphs, and offer several follow-up questions.

Copyright
References
Hide All
[1] Benjamini, I. and Kesten, H. (1996) Distinguishing sceneries by observing the scenery along a random walk path. J. Anal. Math. 69 97135.
[2] Finucane, H., Tamuz, O. and Yaari, Y. (2014) Scenery reconstruction on finite abelian groups. Stochastic Process. Appl. 124 27542770.
[3] Garban, C. and Steif, J. E. (2015) Noise Sensitivity of Boolean Functions and Percolation, Institute of Mathematical Statistics Textbooks, Cambridge University Press.
[4] Hardy, G. H. and Ramanujan, S. (1918) Asymptotic formulæ in combinatory analysis. Proc. London Math. Soc. s2-17 75115.
[5] Krotov, D. S. and Avgustinovich, S. V. (2008) On the number of 1-perfect binary codes: A lower bound. IEEE Trans. Inform. Theor. 54 17601765.
[6] Lindenstrauss, E. (1999) Indistinguishable sceneries. Random Struct. Alg. 14 7186.
[7] van Lint, J. H. (1975) A survey of perfect codes. Rocky Mountain J. Math. 5 199224.
[8] van Lint, J. H. (1998) Introduction to Coding Theory, third edition, Springer.
[9] Matzinger, H. and Rolles, S. W. (2003) Reconstructing a piece of scenery with polynomially many observations. Stochastic Process. Appl. 107 289300.
[10] O'Donnell, R. (2014) Analysis of Boolean Functions, Cambridge University Press.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed