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 2.5·2^n) and large (from 14·3^{n-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^{αn}, 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: B_0 consists of three rationals −1, 0, 1; B_n consists of ordered triples (a, b, c) of elements from B_{n-1} such that a+b+c=0.

Cite: Krotov D.S. , Potapov V.N.
On the Cardinality Spectrum and the Number of Latin Bitrades of Order 3
Problems of Information Transmission. 2019. V.55. N4. P.343-365. DOI: 10.1134/s0032946019040021 WOS Scopus РИНЦ OpenAlex

Original: Кротов Д.С. , Потапов В.Н.
О спектре мощностей и числе латинских битрейдов порядка 3
Проблемы передачи информации. 2019. Т.55. №4. С.52-75. DOI: 10.1134/s0555292319040028 РИНЦ OpenAlex

Dates:

Submitted:	Dec 24, 2018
Accepted:	Nov 12, 2019
Published online:	Jan 24, 2020

Identifiers:

Web of science:	WOS:000520150600002
Scopus:	2-s2.0-85078134798
Elibrary:	43239749
OpenAlex:	W3001040206

Citing:

DB	Citing
Web of science	2
Scopus	2
Elibrary	3
OpenAlex	5

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