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English version:
Journal of Applied and Industrial Mathematics, 2019, 13:4, 717-739

Volume 26, No 4, 2019, P. 74-107

UDC 519.714
K. L. Rychkov
On the perfectness of minimal regular partitions of the edge set of the $n$-dimensional cube

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 ($\pi$-schemes) realizing the linear Boolean functions that depend essentially on $n$ variables for the corresponding values of $n$.
Bibliogr. 16.

Keywords: Boolean function, $\pi$-scheme, regular partition of the edge set of the $n$-dimensional cube, lower complexity bound.

DOI: 10.33048/daio.2019.26.662

Konstantin L. Rychkov 1
1. Sobolev Institute of Mathematics,
4 Koptyug Ave., 630090 Novosibirsk, Russia
e-mail: rychkov@math.nsc.ru

Received June 10, 2019
Revised July 29, 2019
Accepted August 28, 2019

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