5 research outputs found
On the cardinality spectrum and the number of latin bitrades of order 3
By a (latin) unitrade, we call a set of vertices of the Hamming graph that is
intersects with every maximal clique in or 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
with (ternary hypercube) and the growth of the number of such bitrades as
grows. In particular, we determine all possible (up to ) and
large (from ) cardinatities of bitrades and prove that the
cardinality of a bitrade is compartible to or modulo (this result
has a treatment in terms of a ternary code of Reed--Muller type). A part of the
results is valid for any . We prove that the number of nonequivalent
bitrades is not less than and is not greater than
, , as .Comment: 18 pp. In Russia