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Sib. Èlektron. Mat. Izv., 2010, Volume 7, Pages 372–382 (Mi semr248)  

This article is cited in 5 scientific papers (total in 5 papers)

Research papers

On perfect colorings of Boolean $n$-cube and correlation immune functions with small density

V. N. Potapov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: A coloring of Boolean $n$-cube is called perfect if, for every vertex, the collection of colors of its neighbors depends only on its own color. Parameters of a perfect coloring are given by an array. A Boolean function is called correlation immune of degree $n-m$ if it takes the value $1$ equal number of times on any $m$-face of Boolean $n$-cube. It is proved that Boolean function $\chi^S$ ($S\subset E^n$) is a perfect coloring if it satisfies the equality $\rho(S)=1-\frac{n}{2(1+\mathrm{cor}(S))}$, where $\mathrm{cor}(S)$ is the maximum degree of the correlation immune of $\chi^S$ and $\rho(S)=|S|/2^n$.
It is offered a straightforward concatenative construction for a perfect coloring of Boolean $n$-cube with array $ (
\begin{array}{cc} 0 & k(2^s-1)
k & k(2^s-2)
\end{array}
)$. This construction provides a new lower bound on the number of such perfect colorings. Also we give an upper bound for this number. We find the cardinality of the minimal component of perfect coloring with these parameters and prove that any minimal component of such perfect coloring is linear.

Keywords: hypercube, perfect coloring, perfect code, MDS code, correlation-immune function, component.

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Document Type: Article
UDC: 519.14, 519.174.7
MSC: 05B15, 05C15
Received July 30, 2010, published November 4, 2010

Citation: V. N. Potapov, “On perfect colorings of Boolean $n$-cube and correlation immune functions with small density”, Sib. Èlektron. Mat. Izv., 7 (2010), 372–382

Citation in format AMSBIB
\Bibitem{Pot10}
\by V.~N.~Potapov
\paper On perfect colorings of Boolean $n$-cube and correlation immune functions with small density
\jour Sib. \`Elektron. Mat. Izv.
\yr 2010
\vol 7
\pages 372--382
\mathnet{http://mi.mathnet.ru/semr248}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. N. Potapov, “O sovershennykh 2-raskraskakh $q$-znachnogo giperkuba”, PDM, 2011, prilozhenie № 4, 18–20  mathnet
    2. V. N. Potapov, “Cardinality spectra of components of correlation immune functions, bent functions, perfect colorings, and codes”, Problems Inform. Transmission, 48:1 (2012), 47–55  mathnet  crossref  isi
    3. Potapov V.N., “On perfect 2-colorings of the q-ary n-cube”, Discrete Math, 312:6 (2012), 1269–1272  crossref  mathscinet  zmath  isi  elib
    4. K. V. Vorobev, “Kratnye sovershennye kody v giperkube”, Diskretn. analiz i issled. oper., 19:4 (2012), 60–65  mathnet  mathscinet
    5. V. N. Potapov, “O bulevykh funktsiyakh, pochti uravnoveshennykh v granyakh”, PDM. Prilozhenie, 2012, no. 5, 23–25  mathnet
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