On $k$-transitivity conditions of a product of regular permutation groups

The paper analyses the product of $m$ regular permutation groups ${G_1}\cdot\ldots\cdot{G_{m}}$, where $m \geq 2 $ is natural number. Each of regular permutation groups is the subgroup of symmetric permutation group $S(\Omega)$ of degree $|\Omega|$ for the set $\Omega$. M. M. Glukhov proved that for $k=2$ and $m=2$, $2$-transitivity of the product ${G_1}\cdot{G_{2}}$ is equivalent to the absence of zeros in the corresponding square matrix with number of rows and columns equal to $|\Omega|-1$. Also by M. M. Glukhov necessary conditions of $2$-transitivity of such product of regular permutation groups are given.
In this paper, we consider the general case for any natural $m$ and $k$ such that $m \geq 2 $ and $k \geq 2 $. It is proved that $k$-transitivity of product of regular permutation groups ${G_1}\cdot\ldots\cdot{G_{m}}$ is equivalent to the absence of zeros in the square matrix with number of rows and columns equal to $(|\Omega | - 1)!/(|\Omega | - k)!$. We obtain correlation between the number of arcs corresponding to this matrix and a natural number $ l $ such that the product $(PsQt)^{l}$ is $2$-transitive, where $P,Q \subseteq S(\Omega )$ are some regular permutation groups and permutation $st$ is $(|\Omega | - 1)$-loop. We provide an example of the building of AES ciphers such that their round transformation are $ k $-transitive on a number of rounds.