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Problems of Information Transmission

, Volume 48, Issue 2, pp 185–192 | Cite as

Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube

  • K. Yu. Gorbunov
  • A. V. Seliverstov
  • V. A. Lyubetsky
Large Systems

Abstract

In a space of dimension 30 we find a pair of parallel hyperplanes, uniquely determined by vertices of a unit cube lying on them, such that strictly between the hyperplanes there are no vertices of the cube, though there are integer points. A similar two-sided example is constructed in dimension 37. We consider possible locations of empty quadrics with respect to vertices of the cube, which is a particular case of a discrete optimization problem for a quadratic polynomial on the set of vertices of the cube. We demonstrate existence of a large number of pairs of parallel hyperplanes such that each pair contains a large number of points of a prescribed set.

Keywords

Information Transmission Dynamic Programming Algorithm Quadratic Polynomial Integer Point Great Common Divisor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • K. Yu. Gorbunov
    • 1
  • A. V. Seliverstov
    • 1
  • V. A. Lyubetsky
    • 1
  1. 1.Kharkevich Institute for Information Transmission ProblemsRASMoscowRussia

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