On the cardinality spectrum and the number of latin bitrades of order 3

Abstract

By a (latin) unitrade, we call a set of vertices of the Hamming graph that is intersects with every maximal clique in $0$ or $2$ vertices. A bitrade is a bipartite unitrade, that is, a unitrade splittable into two independent sets. We study the cardinality spectrum of the 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 (up to $2.5\cdot 2^n$) and large (from $14\cdot 3^{n-3}$) cardinatities of bitrades and prove that the cardinality of a bitrade is compartible to $0$ or $2^n$ modulo $3$ (this result has a treatment in terms of a ternary code of Reed--Muller type). A part of the results is valid for any $k$. We prove that the number of nonequivalent bitrades is not less than $2^{(2/3-o(1))n}$ and is not greater than $2^{\alpha^n}$, $\alpha<2$, as $n\to\infty$.