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. GorbunovEmail author
  • A. V. Seliverstov
  • V. A. Lyubetsky
Large Systems


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.


Information Transmission Dynamic Programming Algorithm Quadratic Polynomial Integer Point Great Common Divisor 
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Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

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

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