$ k$-homogeneous Latin Squares, their transversals and condition of orthogonality

We consider the $k$-homogeneous Latin Squares, i.e. Latin Squares of order $kn$ with elements from $\{0,\dots,kn-1\}$ such that reducing modulo $n$ gives $(kn \times kn)$-matrix consisting of $k^2$ identical Latin Squares of order $n$. Some characteristics of transversals for $k$-homogeneous Latin Squares are described. Sufficient condition of orthogonality is presented.