On the Cardinality Spectrum and the Number of Latin Bitrades of Order 3
- Authors: Krotov D.S.1, Potapov V.N.1
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Affiliations:
- Sobolev Institute of Mathematics
- Issue: Vol 55, No 4 (2019)
- Pages: 343-365
- Section: Coding Theory
- URL: https://journal-vniispk.ru/0032-9460/article/view/166640
- DOI: https://doi.org/10.1134/S0032946019040021
- ID: 166640
Cite item
Abstract
By a (Latin) unitrade of order k, we call a subset of vertices of the Hamming graph H(n, k) that intersects any maximal clique at either 0 or 2 vertices. A bitrade is a bipartite unitrade, i.e., a unitrade that can be split into two independent subsets. We study the cardinality spectrum of bitrades in the Hamming graph H(n, k) with k = 3 (ternary hypercube) and the growth of the number of such bitrades as n grows. In particular, we determine all possible small (up to 2.5·2n) and large (from 14·3n−3) cardinalities of bitrades of dimension n and prove that the cardinality of a bitrade takes only values equivalent to 0 or 2n modulo 3 (this result can be treated in terms of a ternary Reed-Muller type code). A part of the results are valid for an arbitrary k. For k = 3 and n → ∞ we prove that the number of nonequivalent bitrades is not less than 2(2/3−o(1))n and not greater than \(2^{\alpha^n}\), α < 2 (the growth order of the double logarithm of this number remains unknown). Alternatively, the studied set Bn of bitrades of order 3 can be defined as follows: B0 consists of three rationals - 1, 0, 1; Bn consists of ordered triples (a, b, c) of elements from Bn−1 such that a + b + c = 0.
About the authors
D. S. Krotov
Sobolev Institute of Mathematics
Author for correspondence.
Email: krotov@math.nsc.ru
Russian Federation, Novosibirsk
V. N. Potapov
Sobolev Institute of Mathematics
Email: krotov@math.nsc.ru
Russian Federation, Novosibirsk
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