Abstract
We prove that, for n equal to 3, 5, and a power of 2, every minimal partition of the edge set of the n-dimensional cube is perfect. As a consequence, we obtain some description of the classes of all minimal parallel-serial contact schemes (π-schemes) realizing the linear Boolean functions that depend essentially on n variables for the corresponding values of n.
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Funding
The author was supported by Program no. 1.1.I of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences (project no. 0314-2019-0001).
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Russian Text © The Author(s), 2019, published in Diskretnyi Analiz i Issledovanie Operatsii, 2019, Vol. 26, No. 4, pp. 63–96.
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Rychkov, K.L. On the Perfectness of Minimal Regular Partitions of the Edge Set of the n-Dimensional Cube. J. Appl. Ind. Math. 13, 717–739 (2019). https://doi.org/10.1134/S1990478919040148
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DOI: https://doi.org/10.1134/S1990478919040148