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Geometry of Translations on a Boolean Cube

Abstract

The operation of Minkowski addition of geometric figures has a discrete analog, addition of subsets of a Boolean cube viewed as a vector space over the two-element field. Subsets of the Boolean cube (or multivariable Boolean functions) form a monoid with respect to this operation. This monoid is of interest in classical discrete analysis as well as in a number of problems related to information theory. We consider several complexity aspects of this monoid, namely structural, algorithmic, and algebraic.

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Correspondence to M. N. Vyalyi or V. K. Leontiev.

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Russian Text © The Author(s), 2019, published in Problemy Peredachi Informatsii, 2019, Vol. 55, No. 2, pp. 58–81.

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Vyalyi, M.N., Leontiev, V.K. Geometry of Translations on a Boolean Cube. Probl Inf Transm 55, 152–173 (2019). https://doi.org/10.1134/S0032946019020042

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Key words

  • Minkowski addition
  • Boolean cube
  • monoid
  • generating elements
  • primitive elements
  • sequence of multiples