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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2016, Volume 13, Pages 888–896
DOI: https://doi.org/10.17377/semi.2016.13.071
(Mi semr721)
 

This article is cited in 1 scientific paper (total in 1 paper)

Discrete mathematics and mathematical cybernetics

On packings of $(n,k)$-products

A. V. Sauskana, Yu. V. Tarannikovb

a Nab. Admirala Tributsa, 37–20, 236006, Kaliningrad, Russia
b Mech. & Math. Department, Lomonosov Moscow State University, 119992, Moscow, Russia
Full-text PDF (149 kB) Citations (1)
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Abstract: An $(n, k)$-product (or simply a product), $n\ge 2k$, is the product of $k$ binomials on the set of $n$ variables; the variables in the product are not repeated. The decomposition of a product is the set of $2^k$ monomials of length $k$ appearing after expanding the brackets in this product. The sum of some products is called a packing if after the decomposition of all products in this sum every monomial appears at most once. The length of the sum of products is the number of products in this sum. A packing is called perfect if every possible monomial of length $k$ appears exactly once. The problem of packings is motivated by the construction of Boolean functions with cryptographically important properties. In the paper we give recursive constructions of packings of products (including perfect ones) and the corresponding recurrence bounds on their length. We give necessary conditions on the parameters $n$ and $k$ for the existence of a perfect packing of $(n, k)$-products. We give the complete solution of the problem of the existence of perfect packings of $(n,k)$-products for $k\le 3$. We find the exact value for the maximal length of a packing of $(n, 2)$-products for any $n$.
Keywords: Packings, combinatorial designs, perfect structures, combinatorial constructions, coding theory, Boolean functions, cryptography, nonlinearity, resiliency, maximal possible nonlinearity, bounds.
Funding agency Grant number
Russian Foundation for Basic Research 16-01-00226_а
The work of the second author is supported by RFBR, grant 16–01–00226.
Received August 22, 2016, published October 24, 2016
Bibliographic databases:
Document Type: Article
UDC: 519.147
MSC: 05B40
Language: English
Citation: A. V. Sauskan, Yu. V. Tarannikov, “On packings of $(n,k)$-products”, Sib. Èlektron. Mat. Izv., 13 (2016), 888–896
Citation in format AMSBIB
\Bibitem{SauTar16}
\by A.~V.~Sauskan, Yu.~V.~Tarannikov
\paper On packings of $(n,k)$-products
\jour Sib. \`Elektron. Mat. Izv.
\yr 2016
\vol 13
\pages 888--896
\mathnet{http://mi.mathnet.ru/semr721}
\crossref{https://doi.org/10.17377/semi.2016.13.071}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000407781100071}
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  • https://www.mathnet.ru/eng/semr/v13/p888
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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