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 B_n of bitrades of order 3 can be defined as follows: B₀ 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.

中文翻译：

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关于基数谱和三阶拉丁双态交易的数量

通过阶为k的（拉丁）单交易，我们称为汉明图H（n，k）的一个顶点子集，该子集在0或2个顶点处与任何最大集团相交。双向交易是一种双向交易，即可以分为两个独立子集的双向交易。我们在k = 3（三元超立方体）的汉明图H（n，k）中研究双向交易的基数谱，并且随着n的增长，双向交易的数量也随之增长。特别地，我们确定维数为n的双交易的所有可能的小基数（最大2.5·2 ⁿ）和大基数（从14·3 ^{n -3}）并证明双向交易的基数仅取等于0或2 ⁿ模3的值（可以用三重Reed-Muller类型代码来对待此结果）。结果的一部分对任意k有效。对于k = 3且n →∞，我们证明了非等价双交易的数量不小于2 ^（2 / ^{3− o（1））n}且不大于\（2 ^ {\ alpha ^ n} \），α <2（此数字的双对数的增长顺序仍然未知）。或者，可以将三阶双交易的研究集B _n定义如下：B ₀由三个有理数组成-1，0，1; B _n由来自B _{n -1}的元素的有序三元组（a，b，c）组成，使得a + b + c = 0。