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 2n+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.
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Original Russian Text Copyright © 2013 Potapov V.N.
The author was supported by the Russian Foundation for Basic Research (Grants 11-01-00997 and 10-01-00616) and the Federal Target Program “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” for 2009–2013 (State Contract 02.740.11.0362).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 54, No. 2, pp. 407–416, March–April, 2013.
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Potapov, V.N. Multidimensional Latin bitrades. Sib Math J 54, 317–324 (2013). https://doi.org/10.1134/S0037446613020146
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DOI: https://doi.org/10.1134/S0037446613020146