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 JIS  Vol.7 No.4 , July 2016
On Addition of Sets in Boolean Space
Abstract: In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operation of sets (namely, Minkowski addition) in Boolean space Bn are presented. Also, sums and multisums of various “classical figures” as: sphere, layer, interval etc. are considered. The obtained results make possible to describe multisums by such characteristics of summands as: the sphere radius, weight of layer, dimension of interval etc. using the methods presented in [2], as well as possible solutions of the equation X+Y=A, where  , are considered. In spite of simplicity of the statement of the problem, complexity of its solutions is obvious at once, when the connection of solutions with constructions of equidistant codes or existence the Hadamard matrices is apparent. The present paper submits certain results (statements) which are to be the ground for next investigations dealing with Minkowski summation operations of sets in Boolean space.
Cite this paper: Leontiev, V. , Movsisyan, G. and Margaryan, Z. (2016) On Addition of Sets in Boolean Space. Journal of Information Security, 7, 232-244. doi: 10.4236/jis.2016.74019.
References

[1]   Leontiev, V.K., Movsisyan, G.L. and Margaryan, Zh.G. (2016) Algebra and Geometry of Sets in Boolean Space. Open Journal of Discrete Mathematics (OJDM), 6, 25-40.

[2]   Movsisyan, G.L. (2013) Dirichlet Regions and Perfect Codes in Additive Channel. Open Journal of Discrete Mathematics (OJDM), 3, 137-142.

[3]   Sachkow, W.N. (1977) Combinatory Methods of Discrete Mathematics. Nauka, Moscow. (In Russian)

[4]   Leontiev, V.K., Movsisyan, G.L. and Osipyan, A. (2014) Classification of the Subsets and the Additive Channels. Open Journal of Discrete Mathematics (OJDM), 4, 67-76.

[5]   Leontiev, V.K. (2001) Selected Problems of Combinatorial Analysis. Bauman Moscow State Technical University, Moscow. (In Russian)

[6]   Leontiev, V.K. (2015) Combinatorics and Information. Moscow Institute of Physics and Technology (MIPT), Moscow. (In Russian)

[7]   Lang, S. (1968) Algebra. Moscow, Mir. (In Russian)

[8]   Nigmatulin, R.G. (1991) Complexity of Boolean Functions. Nauka, Moscow, p. 240. (In Russian)

[9]   Movsisyan, G.L. (1982) Perfect Codes in the Schemes Johnson. Bulletin of MSY, Computing Mathematics and Cybernetics, 1, 64-69. (In Russian)

[10]   Leontiev, V.K., Movsisyan, G.L. and Margaryan, Zh.G. (2012) Constant Weight of Perfect and D-Representable Codes. Proceedings of the Yerevan State University, Physical and Mathematical Sciences, 16-19.

[11]   McWilliams, F.J. and Sloane, N.J.A. (1977) The Theory of Error-Correcting Codes. Parts I and II, North-Holland Publishing Company.

[12]   Delsarte, P. (1973) Four Fundamental Parameters of a Code and Their Combinatorial Significance. Information and Control, 23, 407-438.

 
 
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