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Sibirsk. Mat. Zh., 2013, Volume 54, Number 2, Pages 407–416 (Mi smj2429)  

This article is cited in 3 scientific papers (total in 3 papers)

Multidimensional Latin bitrades

V. N. Potapov

Novosibirsk State University, Novosibirsk, Russia

Abstract: A subset of the $n$-dimensional $k$-valued hypercube is a unitrade or united bitrade whenever the size of its intersections with the one-dimensional faces of the hypercube takes only the values $0$ and $2$. A unitrade is bipartite or Hamiltonian whenever the corresponding subgraph of the hypercube is bipartite or Hamiltonian. The pair of parts of a bipartite unitrade is an $n$-dimensional Latin bitrade. For the $n$-dimensional ternary hypercube we determine the number of distinct unitrades and obtain an exponential lower bound on the number of inequivalent Latin bitrades. We list all possible $n$-dimensional Latin bitrades of size less than $2^{n+1}$.
A subset of the $n$-dimensional $k$-valued hypercube is a $t$-fold MDS code whenever the size of its intersection with each one-dimensional face of the hypercube is exactly $t$. The symmetric difference of two single MDS codes is a bipartite unitrade. Each component of the corresponding Latin bitrade is a switching component of one of these MDS codes. We study the sizes of the components of MDS codes and the possibility of obtaining Latin bitrades of a size given from MDS codes. Furthermore, each MDS code is shown to embed in a Hamiltonian $2$-fold MDS code.

Keywords: MDS code, Latin bitrade, unitrade, component.

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English version:
Siberian Mathematical Journal, 2013, 54:2, 317–324

Bibliographic databases:

UDC: 519.14
Received: 17.03.2012

Citation: V. N. Potapov, “Multidimensional Latin bitrades”, Sibirsk. Mat. Zh., 54:2 (2013), 407–416; Siberian Math. J., 54:2 (2013), 317–324

Citation in format AMSBIB
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\by V.~N.~Potapov
\paper Multidimensional Latin bitrades
\jour Sibirsk. Mat. Zh.
\yr 2013
\vol 54
\issue 2
\pages 407--416
\mathnet{http://mi.mathnet.ru/smj2429}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3088605}
\transl
\jour Siberian Math. J.
\yr 2013
\vol 54
\issue 2
\pages 317--324
\crossref{https://doi.org/10.1134/S0037446613020146}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84876447423}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. K. V. Vorob'ev, D. S. Krotov, “Bounds on the cardinality of a minimal $1$-perfect bitrade in the Hamming graph”, J. Appl. Industr. Math., 9:1 (2015), 141–146  mathnet  crossref  mathscinet
    2. D. S. Krotov, I. Yu. Mogilnykh, V. N. Potapov, “To the theory of $q$-ary Steiner and other-type trades”, Discrete Math., 339:3 (2016), 1150–1157  crossref  mathscinet  zmath  isi  elib  scopus
    3. D. S. Krotov, “On the gaps of the spectrum of volumes of trades”, J. Comb Des., 26:3 (2018), 119–126  crossref  mathscinet  zmath  isi  scopus
  • Сибирский математический журнал Siberian Mathematical Journal
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