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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References
Leontiev, V., Movsisyan, G., and Margaryan, Zh., On Addition of Sets in Boolean Space, J. Inf. Secur., 2016, vol. 7, no. 4, pp. 232–244.
Leontiev, V., Movsisyan, G., and Margaryan, Zh., Algebra and Geometry of Sets in Boolean Space, Open J. Discrete Math., 2016, vol. 6, no. 2. P. 25–40.
Leont’ev, V.K., Movsisyan, G.L., and Osipyan, A.A., Classification of Subsets of B n and Additive Channels, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2014, no. 5, pp. 23–29 [Moscow Univ. Math. Bull. (Engl. Transl.), 2014, vol. 69, no. 5, pp. 198–204].
Leontiev, V., Movsisyan, G., Osipyan, A., and Margaryan, Zh., On the Matrix and Additive Communication Channels, J. Inf. Secur., 2014, vol. 5, no. 4, pp. 178–191.
Sipser, M., Introduction to the Theory of Computation, Boston, MA: Thomson Course Technology, 2006, 2nd ed.
Kitaev, A.Yu., Shen, A.H., and Vyalyi, M.N., Klassicheskie i kvantovye vychisleniya, Moscow: MCCME, 1999. Translated under the title Classical and Quantum Computation, Providence, R.I.: Amer. Math. Soc., 2002.
Leont’ev, V.K., On the Faces of an n-Dimensional Cube, Zh. Vychisl. Mat. Mat. Fiz., 2008, vol. 48, no. 6, pp. 1126–1139 [Comput. Math. Math. Phys. (Engl. Transl.), 2008, vol. 48, no. 6, pp. 1063–1075].
Guruswami, V., Micciancio, D., and Regev, O., The Complexity of the Covering Radius Problem on Lattices and Codes, in Proc. 19th IEEE Annual Conf. on Computational Complexity (CCC’04), Amherst, MA, USA, June 21–24, 2004, Washington, DC, USA: IEEE Comput. Soc., 2004, pp. 161–173.
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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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DOI: https://doi.org/10.1134/S0032946019020042